Statistical Inference and Hypothesis Testing

This material builds a bridge from the normal distribution into the two big tools of statistics: estimation and hypothesis testing. We first revisit the normal distribution and the Z score, then see how a single sample can stand in for an entire population, how the central limit theorem lets us treat sample means as normal, how to build confidence intervals for a population mean, and finally how hypothesis testing works — the logic of null and alternative hypotheses and the two kinds of error that every test must manage.

The lecture moves along one continuous path. It starts with the machinery you already know (the normal distribution, the Z score, and the normal tables), then asks a bigger question: how can a sample of 30 people speak for a population of thousands? That question leads to the sampling distribution of the sample mean and the central limit theorem, which in turn justify the two tools of the lecture — estimating an unknown mean with a confidence interval, and deciding between two competing claims with a hypothesis test. The coin-tossing game that opens the hypothesis-testing part is worth remembering: it is the same logic that appears in courtrooms, grading rooms, and research labs, and it will reappear in every later lecture on testing procedures.

3.1 The Normal Distribution and the Z-Score: A Recap

3.1.1 Why the normal distribution matters

The normal distribution is one of the most important and fascinating continuous distributions we have. Beyond being a theoretical object, it is widely used in practical problems and in companies, in many different facets. One everyday example comes from teaching itself: when a large class has to be graded, the normal distribution is used as a tool to classify the number of grades — to decide how many students fall into each grade band. That is only one of many applications.

The shape explains why it keeps appearing. If you plot the normal distribution, you get a smooth, symmetric, bell-shaped curve: most of the data sits close to the middle, and values farther from the middle become rarer and rarer, tailing off on both sides in exactly the same way. Height, exam scores, measurement errors in an instrument, the weight of a packaged product, and the monthly return of a stock index are all examples of quantities that are often at least roughly normal. When a process adds up many small independent influences — and much of business and engineering data does — the result tends to look like this bell.

Real-world: grading systems for large classes, plus countless business and engineering problems, rely on the normal distribution — from setting grade bands and quality-control limits on a production line to risk calculations in finance.

3.1.2 Probabilities from normal tables instead of calculus

Earlier we saw how to compute probabilities with the help of normal tables. These tables are used extensively whenever we need probabilities for normal random variables. They made our life easy: instead of doing the calculus — integrating the normal density function and then extracting probabilities from the integral — we read the probability straight from a table. Nobody in this course computes probabilities by integrating complicated functions.

To see what the tables are doing, the normal density for a random variable $X$ with mean $\mu$ and standard deviation $\sigma$ is

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}(-\infty <x<\infty ),$$

where $e$ is Euler's number (about 2.718) and $\pi$ is about 3.142. A probability such as $P(a\le X\le b)$ is the area under this curve between $a$ and $b$ — and computing that area by hand requires calculus. The normal tables skip all of it: the areas have already been worked out for one standard curve, and we simply look them up.

The tables are built cumulatively: the probability accumulates from the far left of the distribution up to each value. In other words, each table entry reports the area under the curve to the left of a given point. That is why a "probability between two values" is always read as a difference of two cumulative entries — you take the left area up to the upper value and subtract the left area up to the lower value. We will use exactly this subtraction in the worked example of Section 3.3.

3.1.3 Standardizing the variable: the Z score

To avoid the calculus entirely, we introduced the idea of standardizing the variable. Any normal random variable can be converted into its standard normal variable — the standard normal distribution, which is the normal distribution with mean 0 and standard deviation 1. This object goes by many names: the standard normal variable, the Z score, the Z tables, or the Z test. People use these words interchangeably whenever they talk about the standard normal distribution, and the notation we use is $Z$.

The transformation itself is the basic formula:

$$Z=\frac{X-\mu }{\sigma }$$

Here $X$ is any normal random variable, $\mu$ (mu) is the population mean, $\sigma$ (sigma) is the population standard deviation, and $Z$ is the resulting standard normal variable.

The formula is easy to verify at the two most important points. If $X$ equals the mean, then

$$Z=\frac{\mu -\mu }{\sigma }=\frac{0}{\sigma }=0.$$

If $X$ sits one standard deviation above the mean, so $X=\mu +\sigma$, then

$$Z=\frac{(\mu +\sigma )-\mu }{\sigma }=\frac{\sigma }{\sigma }=1.$$

So $Z$ can be read as a distance: the number of standard deviations that the value $X$ lies away from the mean $\mu$. A value two standard deviations below the mean gives $Z=-2$; a value half a standard deviation above the mean gives $Z=0.5$. Every normal distribution, whatever its $\mu$ and $\sigma$, is converted by this formula into the same standard normal distribution with mean 0 and standard deviation 1 — and it is the tables for that one standard distribution that we read everywhere.

Visual intuition. Picture the standard normal curve: the horizontal axis carries the Z values (unitless, from about −3.5 to +3.5 in practice), and the vertical axis carries the density (a height, not a probability). The curve is a symmetric bell peaking at $Z=0$ (the mean), where the height is about 0.4, and it falls off toward zero in both tails. The total area under the curve is exactly 1 — the total probability. Landmarks: the inflection points (where the curve stops curving down and starts turning outward) sit at $Z=\pm 1$, and about 68 percent of the area lies between them, which is the empirical rule we will meet again in Section 3.3.5. The takeaway: any normal problem, drawn in any units, becomes this one fixed picture once standardized.

3.1.4 Z-scores for spotting outliers

One interesting application of the Z score is anticipating outliers in a dataset. Box plots and other pictorial representations are one way to extract outliers; the Z score is another tool that helps us figure out whether any outliers exist in our data.

The working rule of thumb follows from the bell curve's landmarks: a value whose Z is beyond $\pm 2$ (outside about 95 percent of a normal population) — and especially beyond $\pm 3$ (outside about 99 percent) — is a candidate outlier worth inspecting. A box plot flags the same observations from the raw data (the points beyond the whiskers); the Z score reaches the same verdict on a common scale, which is convenient when variables have different units and cannot be compared directly.

Note that detecting an outlier is a separate question from what that outlier means: whether it carries meaningful hidden information depends on the nature of the dataset, and that debate comes later. An outlier could be a data-entry mistake, a genuine extreme observation, or the most interesting record in the file — the Z score only tells you it is unusual, not what to do about it.

Real-world: outlier screening with Z scores and box plots is routine in data analysis before any modeling work begins — salary audits looking for unusual payments, sensor readings that might be faulty, and fraud-screening where one transaction sits many standard deviations above the typical amount.

3.1.5 Z as the foundation for T, chi-square, and F

From here onward, estimation and hypothesis testing lean on the normal distribution. Depending on the context, later we will use the t distribution, the chi-square distribution, or the F distribution. The underlying point is that once you understand the Z distribution thoroughly, the rest behave in a very similar way — you simply pick the respective distribution and read its tables. The textbook covers these distributions in chapter 3.

The family logic is worth stating once: the Z distribution is the standard normal. The t distribution is a close cousin used when the population standard deviation is unknown and the sample is small; the chi-square and F distributions are built from sums of squared normal values and are used for variances, goodness-of-fit, and analysis of variance. In each case the habit is the same — standardize the quantity of interest, choose the right table, read the probability. If you master the reading of one table, you have learned the reading of them all.

Real-world & domain: this is where the normal distribution earns its keep outside the classroom. In quality control, control charts flag a process as "out of control" when a sample statistic moves beyond $\pm 3$ standard errors — the 3-sigma rule that traces straight back to the bell curve. In finance, value-at-risk calculations estimate the worst loss a portfolio might face using normal quantiles. In education, the grading example from 3.1.1 is itself the applied version: the normal curve decides how many students fall into each grade band. The same curve, the same tables, different industries.

3.2 Statistical Inference: From Samples to Populations

3.2.1 Population parameters versus sample statistics

Statistical inference is a fascinating area of statistics, and ultimately it is the point of everything: whatever analysis you do on a given dataset, you must be able to infer something and give the manager some insightful information. The foundation is the distinction between population and sample. The population is the entire group of interest; the sample is the subset we actually measure. Population characteristics are called parameters; sample characteristics are called statistics. For example, the population mean $\mu$, the population standard deviation $\sigma$, and the population proportion are parameters; their sample counterparts — the sample mean $\bar{x}$, the sample standard deviation $s$, and the sample proportion $\hat{p}$ — are statistics.

The name is worth saying out loud once: an "estimate" is a sample value standing in for an unknown population value. $\bar{x}$ estimates $\mu$, $s$ estimates $\sigma$, and $\hat{p}$ estimates the population proportion $p$.

There are many situations where getting information about the whole population is highly difficult, time consuming, or costly. When the population parameters are unknown, we take a sample of observations and generalize the pattern to the whole population. And here common sense must be admitted: once we come down from the population level to the sample level, committing an error is inevitable. No sample is a perfect photograph of the population — the best we can promise is a well-chosen sample that gets close, with the size of the likely error quantified. That quantification is exactly what the sampling distribution (Section 3.3) and confidence intervals (Section 3.5) provide.

3.2.2 The managers example

The running example: finding the profile of a set of managers, around 2500 of them, and in some companies that number rises much higher. The characteristics of interest are the mean salaries and the proportion of managers who have completed the company's training program. If we consider the entire set of 2500 managers, the characteristics we compute are the population parameters — but that information may not be readily available in the databases, and collecting it for everyone may be impractical in terms of time and cost.

So we take a sample of 30 managers and compute the same characteristics: the sample mean $\bar{x}$, the sample standard deviation $s$, and the sample proportion $\hat{p}$. We then use these to address the population level. The open question — how adequate and reliable sample information really is — is explored later; the full worked data for the 30 managers is in the textbook, worth going through when you have time.

Concretely: with one random sample of 30 managers you might get $\bar{x}=\text{\$52,000}$; with a second sample of 30 (a different set of names) you might get $\bar{x}=\text{\$51,400}$. Both are perfectly valid samples; neither equals the unknown $\mu$ exactly, and both differ from each other. It is the spread of these possible sample means — not the spread of the 2500 salaries — that determines how far the sample mean is likely to land from the truth.

Real-world: companies routinely estimate workforce characteristics — salaries, training completion — from samples of employees rather than from surveys of everyone. The same logic runs through market research (surveying 1000 households to describe a national market), political polling (a few thousand voters standing in for millions), and quality control (inspecting a few parts to certify a whole production batch).

3.2.3 Why we accept error and try to minimize it

Because we drop from the population level to the sample level, error is inevitable. The modern view of quality is not about eliminating error; it is about reducing error. We admit that we may commit a mistake, and we try to minimize it.

This philosophy — quality as error reduction, not error elimination — is not a slogan. It is the working principle of statistical design: larger samples shrink sampling error, random sampling removes bias, and the formulas in the coming sections put a number on the remaining uncertainty. When we later meet hypothesis testing, the same mindset reappears in its sharpest form: the two possible mistakes (type 1 and type 2 errors, Section 3.7) cannot be made to vanish; the craft is choosing which one to suppress and how far to push it down.

Real-world & domain: the "reduce, don't eliminate" view of quality is the modern quality-management position that drives statistical process control in manufacturing, where the target is never "zero defects" but "defects reduced to a known, acceptable rate and monitored continuously." It also shapes the reporting culture of data-driven business: a manager who asks for certainty is asking for the wrong thing — the right question is "how big is the error, and is it small enough for this decision?"

3.3 The Sampling Distribution of the Sample Mean and the Central Limit Theorem

3.3.1 The sample mean is a random variable

Since the sample mean $\bar{x}$ is a random variable, a probability distribution is associated with it: the sampling distribution of $\bar{x}$. The sampling distribution of $\bar{x}$ is the probability distribution of all possible sample means that we would see if we drew every possible random sample of size $n$ and computed $\bar{x}$ from each one. It has a mean, a standard deviation, and a shape — just like any distribution.

The key visual from this part is a figure that observes the pattern of the sample mean while the sample size changes. The parent distributions shown are a uniform distribution, a bimodal distribution, and an exponential distribution. If we keep the sample size $n$ large, the sample mean can be approximated with a normal distribution, without any doubt. This is a very fundamental result.

The displayed variance of $\bar{x}$ also appeared; its formula carried a typographical error that was corrected on the screen. The correct statement is the pair of formulas

$$E(\bar{x})=\mu \text{and}\text{Var}(\bar{x})=\frac{{\sigma }^2}{n}.$$

The first says that the mean of the sampling distribution of $\bar{x}$ is the population mean — on average, the sample mean aims at the truth, so $\bar{x}$ is an unbiased estimator of $\mu$. The second says that the variance of the sample mean shrinks as the sample size grows. The standard deviation of $\bar{x}$ — called the standard error of the mean — is $\sigma /\sqrt{n}$, the value that will appear in every formula in this lecture.

Why ${\sigma }^2/n$? Because the sample mean is an average of independent observations, the variances combine in a manageable way:

$$\begin{array}{rlrl}\text{Var}(\bar{x})& =\text{Var}\left(\frac{1}{n}\sum _{i=1}^n{X}_i\right)& & \text{definition of the sample mean}\\ & =\frac{1}{n^2}\text{Var}\left(\sum _{i=1}^n{X}_i\right)& & \text{Var}(cY)=c^2\text{Var}(Y)\text{ for any constant }c\\ & =\frac{1}{n^2}\sum _{i=1}^n\text{Var}(X_i)& & \text{independent observations: variances add}\\ & =\frac{1}{n^2}\cdot n{\sigma }^2& & \text{each }{X}_i\text{ has variance }{\sigma }^2\\ & =\frac{{\sigma }^2}{n}& & \text{cancel one factor of }n\end{array}$$

Taking the square root gives the standard error $\sigma /\sqrt{n}$. Two consequences worth remembering: the sample mean is more precise than a single observation (its variance is divided by $n$), and precision grows slowly — to halve the standard error you must quadruple the sample size, because the square root only shrinks as $\sqrt{n}$.

Visual intuition. The lecture's figure shows three columns, one per parent population. Column 1 is a uniform distribution (flat, every value equally likely, like a single die roll). Column 2 is a bimodal distribution (two separate humps, like a two-lobed "rabbit-eared" curve). Column 3 is an exponential distribution (sharply skewed right, a steep peak near zero with a long tail). Each column shows the shape of the parent distribution on top, and below it the shape of the sampling distribution for samples of size $n=2$, $n=5$, and $n=30$. With $n=2$ the three sampling distributions still look nothing like each other and nothing like a bell; with $n=5$ the uniform and bimodal columns are already turning bell-shaped, while the exponential still drags to the right; by $n=30$ all three columns are nearly normal. The one-sentence takeaway: sample means forget their parents — the shape of $\bar{x}$'s distribution is decided by $n$, not by the population.

3.3.2 The central limit theorem

The statement above is the central limit theorem (CLT): if $n$ is sufficiently large, the sample mean is approximately normally distributed, irrespective of whatever the parent distribution is — uniform, bimodal, exponential, or anything else. It is a very fundamental theorem that one must know. Because of it, you can use the normal tables whenever the sample mean is in question.

Convention on sample size: many textbooks treat $n$ greater than 30 — 30 or 40 is acceptable — as a large sample. For $n>30$ you can safely use the z distribution, or the normal tables, without any doubt.

3.3.3 The z transformation for sample means

For a single random variable X, the transformation was $Z=(X-\mu )/\sigma$. When we move from single observations to sample means, the transformation changes to:

$$z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$$

Here $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, $n$ is the sample size, and $\sigma /\sqrt{n}$ is the standard deviation of the sampling distribution of $\bar{x}$ — how much sample means spread around the population mean. Remember this particular formula; it is the one that lets you use the normal tables for sample means.

The only structural difference from the single-value version is the denominator: instead of $\sigma$ (the spread of individual values), we divide by $\sigma /\sqrt{n}$ (the spread of sample means). The sample mean's variability is smaller than the data's variability, so the z statistic for a sample mean is inflated relative to the single-value version — a small gap between $\bar{x}$ and $\mu$ can be very surprising when $n$ is large. That is the mechanism by which bigger samples produce sharper conclusions: the same observed gap yields a larger $|z|$, which corresponds to a more extreme, less likely position in the standard normal.

3.3.4 Worked example: probability that $\bar{x}$ is within $\pm 5$ of $\mu$

This is textbook problem Chapter 7, page 326, problem 19. The setup: a population has mean 200 and standard deviation 50. Suppose a sample of size 100 is selected, and $\bar{x}$ is used to estimate $\mu$. What is the probability that the sample mean will be within plus or minus 5 of the population mean?

3.3.5 The empirical rule

The answer matches the empirical rule: one-sigma limits hold about 68 percent of the observations, two-sigma limits hold about 95 percent, and three-sigma limits hold about 99 percent.

$$P(\mu -\sigma \le X\le \mu +\sigma )\approx 0.68,P(\mu -2\sigma \le X\le \mu +2\sigma )\approx 0.95,P(\mu -3\sigma \le X\le \mu +3\sigma )\approx 0.99$$

The empirical rule is a simple and famous fact about the normal distribution: 68-95-99.

Visual intuition. Picture the bell curve again, with the horizontal axis in $\sigma$ units around $\mu$. Mark the points $\mu \pm 1\sigma$: the region between them is the central band containing about 68 percent of the area — the tall middle of the bell. Widen to $\mu \pm 2\sigma$: about 95 percent of the area, leaving only 2.5 percent in each tail. Widen to $\mu \pm 3\sigma$: about 99 percent, with the tiny remaining slivers beyond. The landmarks read like a ruler: the height of the curve where the band stops tells you nothing directly — it is the area inside the band that is the probability. The one-sentence takeaway: a value more than 2 standard deviations from the mean is uncommon (about 5 percent of the population), and more than 3 is very uncommon (about 1 percent) — the same numbers that make the ±2 and ±3 Z-score outlier rules of Section 3.1.4 work.

The rule rounds the exact normal probabilities: the precise values are 0.6827 (one sigma), 0.9545 (two sigma), and 0.9973 (three sigma) — and the exact 95 percent multiplier is $z=1.96$, not $z=2$. The rounded 68-95-99 form is what everyone quotes for quick interpretation; the table values are what you compute with.

Recap + bridge: The central limit theorem turned the random-variable $\bar{x}$ into a calculable object — nearly normal with mean $\mu$ and standard error $\sigma /\sqrt{n}$ — and the worked example turned that into numbers (0.6826). With probabilities about the sample mean in hand, the next step is the natural one: stop asking "how often is $\bar{x}$ within 5 of $\mu$?" and start asking "which numbers are plausible values of $\mu$?" — that is, point and interval estimation, and then confidence intervals.

Real-world & domain: the CLT is the silent engine behind nearly every published poll and every margin of error in the news — the "±3 percent" attached to an opinion poll is built from $\sigma /\sqrt{n}$ and the normal tables. In manufacturing, control charts plot successive sample means and flag any mean beyond $\mu \pm 3\sigma /\sqrt{n}$, the 99 percent CLT band, as a signal that the process has changed. Wherever someone quotes an average with a confidence interval — customer satisfaction scores, quality metrics, election predictions — the sampling distribution of the sample mean is doing the work.

3.4 Point Estimates and Interval Estimates

3.4.1 Point estimate

When a population parameter such as $\mu$ is unknown, we replace it with sample information. There are two ways to represent that replacement. The first is with a single number: we replace $\mu$ with $\bar{x}$. Reporting a single number is a point estimate.

The everyday version — the professor's own: before results are out, estimate your total marks in the comprehensive exam. A confident student says "80", another says "70", another says "30" — each is reporting a point estimate, one number standing for the unknown value. Notice what these three guesses share: they are single numbers, easy to state, easy to compare, and all of them are almost certainly wrong in the exact sense that none equals the true score. A point estimate's virtue is simplicity; its weakness is that it carries no information about how wrong it might be.

The formal picture: the point estimate $\bar{x}$ replaces the unknown constant $\mu$. We already know from Section 3.3 that $\bar{x}$ is a random variable centered at $\mu$ with standard error $\sigma /\sqrt{n}$ — so a point estimate is best understood as "a draw from a distribution," and the size of that distribution's spread tells you how far off the single number might be. That link between point estimation and the sampling distribution is the bridge to the interval approach.

3.4.2 Interval estimate

The alternative is to report a range: "I expect marks between 60 and 80", or "between 70 and 80". That is an interval estimate — we estimate the population parameter not with a single number but with an interval.

The interval approach enhances the confidence of expecting: it is generally better to represent the unknown over an interval than with a single value, which is why confidence intervals come next. The trade is visible in the marks example: "between 60 and 80" hedges against the guess being off by a few marks, and "between 70 and 80" is a tighter, more informative range. An interval answers the question the point estimate leaves open: not just "where do I think it is?" but "how far off could I be?" The width of the interval is the honest admission of uncertainty.

Real-world: interval reporting is standard in business and public discourse — polling results are published as "53 percent, with a margin of error of plus or minus 3 points," never as a bare point estimate — because a range conveys both the estimate and its reliability.

3.4.3 Why a 0 to 100 interval is not useful

A clever student could say "I will get marks between 0 and 100" — that interval certainly contains the outcome, but it is useless. The question was raised as: why settle for a confidence level of 95 percent when a level of 100 percent is available? The answer: such large intervals are not precise and they give fuzzy estimates.

This is exactly the motivation for confidence intervals — the trade-off between how confident we are and how precise the interval is. (The derivation of confidence intervals is in the textbook; here we apply the formulas to problems.)

Real-world & domain: the confidence-versus-precision trade-off is negotiated daily in business reporting and public statistics — audit reports, market-share estimates, and election polls all state a margin of error precisely because an unaccompanied point estimate would overstate certainty, and a 0-to-100-style range would say nothing. The skill of reading any published number is really the skill of asking: what is the interval, and at what confidence?

3.5 Confidence Intervals for the Population Mean (Sigma Known)

3.5.1 The confidence interval formula

We now build confidence intervals for the population mean — the context of averages. When the target is the population mean and $\sigma$ is known — meaning the population standard deviation is given in the problem — the confidence interval is:

$$\bar{x}\pm z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$$

The lower limit is $\bar{x}-z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$ and the upper limit is $\bar{x}+z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$.

Here $\bar{x}$ is the sample mean, $\sigma$ the known population standard deviation, $n$ the sample size, and $z_{\alpha /2}$ (z alpha by 2) a value read from the normal table. Previously we replaced $\mu$ with the single number $\bar{x}$; now we estimate $\mu$ over an interval. Remember this expression — it is the confidence interval for the population mean when sigma is known.

Why this particular form? The interval is the point estimate $\bar{x}$ plus or minus a margin of error, and the margin of error is $z_{\alpha /2}$ standard errors. The logic: from Section 3.3, about 95 percent of all sample means lie within $1.96$ standard errors of $\mu$ — that is, within $1.96\cdot \sigma /\sqrt{n}$ of the truth. So for 95 percent of all samples, the interval built by walking $1.96$ standard errors out from the sample's own $\bar{x}$ reaches across to cover $\mu$. The other way to see it: each sample produces its own interval, centered at its own $\bar{x}$, and 95 percent of those intervals succeed in capturing $\mu$.

3.5.2 The z alpha by 2 notation

The only piece to compute beyond the given values is $z_{\alpha /2}$. Its definition: $z_{\alpha /2}$ is the z value providing an area of $\alpha /2$ in the upper tail of the standard normal distribution:

$$P(Z>z_{\alpha /2})=\alpha /2$$

For a 95 percent confidence interval, the confidence level $1-\alpha$ equals 0.95, so $\alpha =0.05$ and $\alpha /2=0.025$. We need $z_{0.025}$: the z value such that the area to its right is 0.025. Because the tables are cumulative, the area to the left of that same value is $1-0.025=0.975$. We search the body of the normal table for 0.975: it sits at 1.9 in the row and 6 in the column, giving $z_{0.025}=1.96$. Equivalently, $P(Z>1.96)=0.025$.

The reading works because of the symmetry of the bell curve: the area in the upper tail (0.025) and the area in the lower tail (0.025) are equal, so the remaining middle 0.95 of the distribution lies between $-z_{0.025}$ and $+z_{0.025}$. The notation $z_{\alpha /2}$ makes the split explicit: the Greek $\alpha$ (alpha) is the total risk of missing, and it is divided by 2 because the misses can happen on either side of the interval.

For quick work, the standard multipliers for the three common confidence levels are:

These are the values you will use again in the B and C parts of the worked problem below, and they are worth memorizing along with the 68-95-99 empirical rule: 1.645, 1.96, 2.576.

3.5.3 Worked example: a 95 percent confidence interval

This is textbook problem Chapter 8, page 355, problem 2. Given: $\bar{x}=32$, $\sigma =6$, and $n=50$. Construct a 95 percent confidence interval for the population mean.

This is a very simple problem: normal tables plus the confidence interval formula. The same problem has further parts (the B and C parts) at other confidence levels; spend time on them for practice, and the full solutions will be shared later.

3.5.4 Z versus t: choosing the distribution by sample size

For large $n$ we use the z test and the z distribution. On a similar note, when $n$ is less than 30, we can make use of the t distribution: everything stays the same except that we read the t distribution tables and replace z with t. Whether we take up the t distribution in detail depends on need; the reason for stressing the z distribution is that once you understand z, the rest follows easily.

The decision rule is a simple threshold, and the machinery is identical on both sides of it:

The conceptual link between them: the t distribution is the z distribution's small-sample cousin — wider (heavier-tailed) than the bell when the sample is tiny, because the sample standard deviation $s$ is itself uncertain, and converging toward the z distribution as $n$ grows. When you have understood z, switching to t is a matter of swapping the table, which is exactly the professor's point about the family of distributions behaving similarly. The reason the threshold convention matters on the exam: the problem statement tells you which table to carry into the formula, and reaching for the wrong table is the classic way to lose marks.

Recap + bridge: The confidence interval $\bar{x}\pm z_{\alpha /2}\sigma /\sqrt{n}$ turned the sampling distribution into a decision tool: a point estimate plus a measured margin of error, interpreted over repeated sampling. The same two ingredients — a sampling distribution and a chosen error risk — will now be rearranged into the second big tool: hypothesis testing, which starts with a coin-tossing game and a question about how much evidence a strange pattern really carries.

3.6 Hypothesis Testing: Motivation from a Coin Tossing Game

3.6.1 The game

The standard opening example for hypothesis testing is a simple game between two friends: coin tossing. If heads appears, the tosser wins; if tails appears, he loses. By agreement, the loser pays one dollar to the winner. Watch how the story builds:

• Toss once: heads. The friend pays 1 dollar.
• Toss twice: heads and heads. The friend pays 2 dollars.
• Toss three times: heads, heads, heads. The friend pays 3 dollars, and we say the winner is a very lucky guy.
• Toss four times: all heads. The friend pays 4 dollars, and again we say the winner is very lucky.
• Toss five times: all heads. The friend pays 5 dollars — and at this point, doubt rises.

The structure of the story is its own lesson. The first few heads are absorbed as luck; the payments are small and the pattern short. Each new head adds one more dollar and one more piece of evidence, and somewhere between the fourth and fifth toss the interpretation flips — the same data (heads) that looked like luck at toss two starts looking like something else at toss five. The game is designed so that the tipping point is felt before it is quantified: nobody needs a formula to get suspicious at five heads in a row.

3.6.2 When luck turns into suspicion

Five heads in a row change the conversation. The question is: is there any possibility that tossing a coin five times gives five heads? Yes — the possibility exists, though it is very small. Bias is one explanation, cheating is another, but the rare pattern itself does not settle the matter. We cannot simply conclude that the coin is biased because of this unusual pattern, yet we also cannot simply ignore it.

Notice the three positions available to a decision maker, and what each gets wrong. Concluding "biased" from five heads risks convicting an innocent coin — the fair coin produces exactly this pattern once in every 32 attempts. Concluding "fair" risks being fleeced — the coin may genuinely be rigged, and the evidence to prove it will only arrive after more losses. Declining to decide is a decision too. Hypothesis testing's answer is to pick a starting assumption, measure how surprising the data are under that assumption, and set a threshold for how much surprise is too much — the framework of the next section.

3.6.3 The student exchange on five heads

The arithmetic is a straight binomial calculation: let $X$ be the number of heads in 5 tosses. For a fair coin, $X$ follows a binomial distribution with $n=5$ and $p=0.5$, and the probability of the specific outcome "all heads" is

$$P(X=5)=\left(\genfrac{}{}{0ex}{}{5}{5}\right)(0.5{)}^5(0.5{)}^0=1\times (0.5{)}^5=\frac{1}{32}=0.03125$$

The numbers for the shorter streaks put the surprise in perspective:

The expected number of heads in 5 tosses is $np=5\times 0.5=2.5$, so five heads sits 2.5 above expectation — far out in the right tail of the distribution. Yet the tail still has area: 3.125 percent of the time, a perfectly fair coin does this. That gap — "surprising but possible" — is precisely the gap hypothesis testing is designed to navigate. Note also the comparison that matters: the probability of "at least one head in 5 tosses" is $1-(0.5{)}^5=31/32\approx 0.969$, very different from the probability of "all five heads" — a reminder always to check whether a question asks about an exact outcome or an "at least" outcome.

Recap + bridge: The coin game turns "I feel suspicious" into a question with a number attached: under the assumption of fairness, five heads has probability 0.03125 — small enough to doubt, large enough to forbid certainty. The next step is to make that intuition formal: name the two competing claims (the null and alternative hypotheses), admit that either decision can be wrong, and classify the two kinds of error. That is the framework of Section 3.7, and the numbers from this game — the 0.03125 pattern probability — will be the seed of the decision rules to come.

Real-world & domain: this exact pattern-recognition logic runs through medical screening and fraud detection, where a rare event under the "normal" assumption triggers an investigation: a cluster of identical transactions flagged by a bank, a test result that lands far outside the healthy range, or a process reading that drifts past its control limits. In each case the question is the coin game's question — how surprising is this pattern if nothing is actually wrong? — and the answer decides whether the anomaly is investigated as evidence or written off as bad luck.

3.7 Hypothesis Testing: Null and Alternative Hypotheses

3.7.1 The criminal and judge example

The classic example for hypothesis testing: a person is caught by the police and produced before a judge. Two statements are in play: the defendant is innocent, or the defendant is a criminal. The police officer's intention is to present evidence that the defendant is a criminal, so that he may be punished. The judge then decides.

For the time being we set aside the question of severity and just fix the framing: assuming that the defendant is innocent is always the safe assumption, so "the defendant is innocent" is the null hypothesis $H_0$, and its negation — "the defendant is a criminal" — is the alternative hypothesis $H_A$.

The roles are worth dwelling on. The null hypothesis $H_0$ is the working assumption — the status quo that stands until evidence knocks it down. The alternative hypothesis $H_A$ is the claim that requires proof; in the courtroom the police must supply the evidence, and the judge's default is release. Hypothesis testing formalizes exactly this asymmetry: the burden of proof sits with the alternative, and the null is let go only when the evidence against it is strong enough.

The judge can pass right decisions: the innocent is released, or the criminal is punished. But the judge can also commit a mistake. Even the judge is going to commit a mistake sometimes — no decision system is free of error.

3.7.2 Type 1 and type 2 errors

The two wrong decisions are the heart of the framework:

• Even though the person is innocent, he may be punished. Rejecting the null hypothesis when $H_0$ is true — treating an innocent as a criminal — is a type 1 error.
• Even though the person is criminal, he may be released. Failing to reject the null when $H_0$ is false — releasing a criminal — is a type 2 error.

The four outcomes of the judge's decision fit the classic two-by-two table:

The definitions are anchored by the truth of $H_0$, not by the court's mood: a type 1 error is always "rejecting a true null," and a type 2 error is always "failing to reject a false null." If you can identify which claim is the null and whether the decision was to reject it, the error type follows mechanically.

Which error is more severe depends on the context, and that can be debated. In the judiciary there is a saying: better to release a thousand criminals than to punish a single innocent. In that view, the type 1 error is more severe than the type 2 error, and the whole framework is built so that the type 1 error gets controlled first. Whatever you decide — accept or reject — you are guaranteed to risk a wrong decision; the entire craft of hypothesis testing is how best to minimize those two errors.

3.7.3 The instructor grading example

A second, personal example: evaluating answers in an examination. The instructor assumes that all the students have written the question correctly — that is the null, just as the defendant was assumed innocent at the start. If the question is correct, giving full marks is right; if the question is wrong, giving zero is right. With a small class of 20 students, each paper can be corrected meticulously. But with a thousand students, in a hurry, mistakes slip in.

The two wrong decisions an instructor can commit mirror the judge's exactly:

• Even though the question is correct, giving zero marks — rejecting $H_0$ when $H_0$ is true — is a type 1 error: a correct question gets punished.
• Even though the question is wrong, giving full marks is a type 2 error.

In this context too, the guiding principle is: a correct question should not be punished. So the type 1 error is the one we fight hardest to minimize.

3.7.4 Minimizing the two errors

The crux of the whole framework: how best to balance the two errors. Given that the type 1 error is treated as the more severe one, the framework is built to keep it as small as possible. At our institute, there is a practical mitigation: the recheck request — a student can request re-evaluation of a graded paper. When a recheck comes in, the type 1 error gets a second look and can be corrected. There is also the honest side: a genuine student who spots a wrongly awarded mark will come back and say "give me zero, you gave me marks by mistake" — that has happened too.

A useful benchmark: if the type 1 error rate is 0.5 — say, 10 of 20 papers wrongly corrected — there is no point in correcting at all. That is the target: the type 1 error should be as close to zero as possible. Everything in the framework — which null, which alternative, which test — is designed around minimizing those two errors, with the type 1 error getting priority.

The benchmark example is deliberately absurd to make the principle vivid: a grader whose "corrections" are wrong half the time — 10 of 20 papers wrongly marked — adds nothing but noise, so the entire enterprise of grading collapses. A test procedure that cannot hold its error near zero is not a test at all. That is why the machinery that follows in later lectures is built the way it is: the choice of null, the choice of alternative, and the choice of test statistic are all made with the type 1 error fixed first, at a small, declared level.

3.7.5 General guidelines for building hypotheses

Three general guidelines close the session and frame the procedures to come:

1. When testing a hypothesis about the value of some parameter $\theta$ (theta), the statement of equality will always be the null hypothesis, and whatever is to be detected or supported is the alternative hypothesis.
2. Since the alternative $H_A$ is the hypothesis we hope to establish, it is to be hoped that the evidence leads us to reject the null hypothesis so that we may accept the alternative.
3. In some situations it is easier to identify the alternative first and then develop the null hypothesis. Many applications of hypothesis testing involve gathering evidence in support of research: in that situation it is often best to begin by writing the alternative, then conclude by rejecting the null so that you can accept the alternative.

Different books phrase these guidelines differently; for this course we follow this convention, and the problems will be addressed accordingly. The next step — the procedures themselves: after constructing the two logical statements, we accept that errors are possible, and the question is how to anticipate and minimize them. That is what the coming material covers, logically and conceptually.

The guidelines are easiest to see in a concrete pair of textbook-style situations. Research framing — a new fuel-injection system is claimed to beat the current 24 miles per gallon: the claim to be supported is $\mu >24$, so the alternative is written first, $H_A:\mu >24$, and the null becomes the complement $H_0:\mu \le 24$. Quality framing — a bottling line is designed to fill at 67.6 fluid ounces: the claim to be detected is any deviation, so $H_A:\mu \ne 67.6$ with $H_0:\mu =67.6$. Notice the pattern both follow and the guideline states: the equality sign lives in the null, and the thing to be detected lives in the alternative.

Recap + bridge: Hypothesis testing frames every decision as a contest between a safe working assumption (the null) and a claim that needs evidence (the alternative); either decision risks a wrong outcome — type 1, rejecting a true null, and type 2, failing to reject a false null — and the framework prioritizes minimizing type 1. The bridge ahead: with the two statements constructed and the two errors named, the coming material supplies the procedures — the test statistic, the threshold, and the rule for when the evidence is enough to reject.

Exam Guidance Summary

Scope and coverage. Revisiting chapters 6 to 8 is what is required at this stage of the course. The textbook, Statistics for Business and Economics, covers the distributions in chapter 3, and the two exam-relevant worked problems appear at chapter 7, page 326, problem 19 and chapter 8, page 355, problem 2. More practice problems with detailed solutions will be shared.

Formulas to remember (each was explicitly flagged during the material):

• The z transformation for a single value: $Z=(X-\mu )/\sigma$.
• The z transformation for the sample mean: $z=(\bar{x}-\mu )/(\sigma /\sqrt{n})$.
• The confidence interval for the population mean with sigma known: $\bar{x}\pm z_{\alpha /2}\sigma /\sqrt{n}$.
• The empirical rule: 68 percent within one sigma, 95 percent within two sigma, 99 percent within three sigma.
• Normal table anchors: $P(Z\le 1)=0.8413$, $P(Z\le -1)=0.1587$, and $z_{0.025}=1.96$ for a 95 percent interval (with $z_{0.05}=1.645$ for 90 percent and $z_{0.005}=2.576$ for 99 percent).
• The pattern probability from the coin game: $(0.5{)}^5=1/32=0.03125$.

Distribution choice. Use the z distribution for large samples (n greater than 30); for small samples (n less than 30) switch to the t distribution — same machinery, t tables instead of z tables.

Hypothesis testing. Understand the motivation (the coin game), the two errors (type 1 = reject a true null, type 2 = fail to reject a false null), and the guidelines for building hypotheses: the statement of equality becomes the null, what is to be detected or supported becomes the alternative, and we hope to reject the null so the alternative can be accepted. The procedures of hypothesis testing are the focus of the coming material and are highly relevant for the upcoming sessions.

Key Industry Applications

• Grading large classes (education): the normal distribution is used as a classification tool to decide how many students fall into each grade band — the everyday opening example of the lecture, and a template for any "bucketize a large population" problem.
• Outlier screening (data analysis): Z scores and box plots are standard tools for detecting outliers in datasets before analysis — Z flags values beyond roughly $\pm 2$ or $\pm 3$ standard deviations, box plots flag points beyond the whiskers; either way, screening runs before any modeling work begins.
• Workforce estimation (human resources): mean salaries and training-program completion are estimated from samples of employees instead of surveying everyone — the managers example of 2500 people with a sample of 30, generalized to any workforce analytics task.
• Quality philosophy (quality management): modern quality management is about reducing error, not eliminating it — the same mindset drives statistical inference, from sampling design to control charts built on the CLT's $\pm 3$ standard-error bands.
• Interval reporting (business reporting, polling): businesses report estimates with confidence intervals rather than single point values to convey precision — "53 percent plus or minus 3 points" instead of a bare number, because the width carries the reliability information.
• Hypothesis testing in research (research practice): research applications gather evidence in support of a claim; the alternative hypothesis is written first and the null is rejected to accept it — the framing used for claims like "the new method is better than the current one" in product research, drug trials, and teaching experiments.
• Judicial and institutional processes (law, administration): the judge example is the classic way to reason about the two kinds of decision errors — punishing the innocent (type 1) versus releasing the criminal (type 2); the recheck-request mechanism is a real institutional response to type 1 error, giving every decision a second look.
• Pattern-vs-luck decisions (finance, medicine, fraud detection): the coin game's logic — is this pattern evidence or bad luck? — drives anomaly detection everywhere, from flagging a cluster of unusual transactions to judging whether an extreme medical test result means the underlying process has changed.