Chapter 8: Hypothesis Testing – Structured Study Notes

Chapter 8: Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical procedure used to evaluate claims about population parameters based on sample data. It provides a structured framework for making decisions under uncertainty.

• Hypothesis: A claim or statement about a property of a population.

• Example: An adult male patient has heart rate readings of 86, 84, 88, 89, 90, 84, 60, 78, 72, and 84. Suppose the national average heart rate for adult males is 72 bpm. The hypothesis could be: "The patient's average heart rate differs from the national average."

Four Parts of a Hypothesis Test

Every hypothesis test consists of four essential steps:

1. Identify and symbolically express the null and alternative hypotheses.

2. Calculate the value of the test statistic using sample data.

3. Assess the test statistic using either the Critical Value Method or the p-value Method.

4. State the conclusion in simple, nontechnical terms.

1. Hypotheses

Null Hypothesis (H0)

The null hypothesis is a statement that the value of a population parameter is equal to some specified value. It is assumed true and tested directly.

• Notation: $H_0$

• Example: $H_0: \mu = 72$

• We "reject" or "fail to reject" $H_0$ based on the evidence.

Alternative Hypothesis (H1 or Ha)

The alternative hypothesis states that the population parameter differs from the value given in the null hypothesis.

• Notation: $H_1$ or $H_a$

• Uses symbols $\neq$, $>$, and $<$

• Example: $H_1: \mu \neq 72$

Forming Hypotheses

• Identify your claim and express it in symbolic form.

• Express the symbolic form that would be true if the original claim is false.

• If you want to support a claim, that claim should become the alternative hypothesis.

• Use $\neq$, $>$, and $<$, but never $=$ for the alternative hypothesis.

Examples of Hypotheses

• The proportion of people ages 45–59 who currently use illegal drugs is 0.3.

• The majority of college students have credit cards.

• The standard deviation of daily rainfall amounts in San Francisco is 0.66 cm.

• The mean weight of plastic discarded by households in one week is less than 1 kg.

2. Test Statistics

Test statistics are used to make decisions about the null hypothesis. They convert sample statistics to a common scale (e.g., z, t, or $\chi^2$).

• Test statistics assume $H_0$ is true.

• Common test statistics:

Parameter	Test Statistic
Mean	$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ or $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$
Proportion	$z = \frac{\hat{p} - p}{\sqrt{pq/n}}$
Standard Deviation	$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

• Example: Heart rate readings: 86, 84, 88, 89, 90, 84, 60, 78, 72, 84. Assume $\mu = 72$.

3. Assess Test Statistic

Critical Value Method

The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis. The critical region depends on:

• The distribution we are using

• The significance level $\alpha$

• The type of hypothesis test

Significance Level ($\alpha$)

• The probability that the test statistic will fall in the critical region when the null hypothesis is true.

• Common values: $\alpha = 0.05$, $\alpha = 0.01$

• Represents the probability of rejecting a true null hypothesis (Type I error).

Types of Hypothesis Tests

• Two-tailed test: Critical region is split evenly between both tails of the curve ($H_1$ uses $\neq$).

• Left-tailed test: Critical region is in the left tail only ($H_1$ uses $<$).

• Right-tailed test: Critical region is in the right tail only ($H_1$ uses $>$).

Critical Value

• A critical value is any value that separates the critical region from the values that do not lead to rejection of the null hypothesis.

• Example: $\alpha = 0.05$

p-value Method

• The p-value is the probability of getting a value of the test statistic that is at least as extreme as the one we have (assuming $H_0$ is true).

• If p-value is very small, we reject the null hypothesis.

• We use the significance level to determine what counts as "very small."

• If p-value $\leq \alpha$, reject $H_0$; if p-value $> \alpha$, fail to reject $H_0$.

Calculating p-values

• For a left-tailed test, the p-value is the area to the left of the test statistic.

• For a right-tailed test, the p-value is the area to the right of the test statistic.

• For a two-tailed test, the p-value is twice the area in the tail "beyond" the test statistic.

4. State the Conclusion

Assessing the test statistic gives an initial conclusion: "Reject" or "Fail to Reject" $H_0$. We never "prove" or "accept" $H_0$; we only say whether we have enough evidence to reject it. The conclusion should be interpreted in the context of the problem and the original claim.

• Example: Our original claim was that the average heart rate was greater than 72, and our initial conclusion was to reject $H_0: \mu = 72$, $H_1: \mu > 72$.

• Example: The mean weight of plastic discarded by households in one week is less than 1 kg. Suppose that we fail to reject $H_0: \mu = 1$ kg, $H_1: \mu < 1$ kg.

Hypothesis Testing – Proportions

Hypothesis tests for proportions are used when the parameter of interest is a population proportion.

• Test statistic for proportion:

$z = \frac{\hat{p} - p}{\sqrt{pq/n}}$

• $\hat{p}$ = sample proportion

• $p$ = claimed population proportion

• $q = 1 - p$

• $n$ = sample size

Example: Critical Value Method

In recent years, a town's arrest rate was 35% for robberies. The arrest rate has been 34% among 30 recent robberies. Test whether there is sufficient evidence to support the claim that her arrest rate is greater than 34%. (Assume $\alpha = 0.05$.)

Example: p-value Method

In a survey of 745 randomly selected adults, 5% said that it is wrong to not report all income on tax returns. Use a 0.05 significance level to test the claim that 75% of adults say that this is wrong.

Hypothesis Testing – Means

Hypothesis tests for means are used when the parameter of interest is a population mean.

• Test statistic for means:

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

• $\bar{x}$ = sample mean

• $\mu$ = claimed population mean

• $\sigma$ = population standard deviation

• $s$ = sample standard deviation

• $n$ = sample size

Example: Critical Value Method

In a simple random sample of 16 college graduates, researchers found that the mean time to degree was 4.8 years and the standard deviation was 1.2 years. Use a 0.05 significance level to test the claim that the mean time to degree for all college students is greater than 4.5 years.

Example: p-value Method

A simple random sample of 1000 scores was collected. Use a 0.05 significance level to test the claim that these FICO scores come from a population with a mean equal to 698. (Data: 714, 765, 753, 864, 849, 773, 698, 759, 819, 802)

Hypothesis Testing – Standard Deviation

Hypothesis tests for standard deviation are used when the parameter of interest is a population standard deviation or variance.

• Test statistic for standard deviation:

$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

• $s$ = sample standard deviation

• $\sigma$ = claimed population standard deviation

• $n$ = sample size

Example: Critical Value Method

Suppose that statistics tests in one professor's class have scores with a standard deviation of 19.4. A recent class of 27 students had test scores with a standard deviation of 13.9. Use a 0.05 significance level to test the claim that this class has less variation than past classes.

Example: p-value Method

The playing times (in seconds) of 16 popular songs are given. Use a 0.05 significance level to test the claim that these songs come from a population with a standard deviation less than 31 seconds. (Data: 44, 243, 213, 216, 295, 295, 295, 257, 246, 243, 213, 216, 295, 295, 295, 257)

Summary Table: Hypothesis Test Types and Test Statistics

Parameter	Test Statistic	Formula	Example Application
Proportion	z	$z = \frac{\hat{p} - p}{\sqrt{pq/n}}$	Testing if the proportion of adults who report all income is 75%
Mean	z or t	$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$	Testing if the mean time to degree is greater than 4.5 years
Standard Deviation	$\chi^2$	$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$	Testing if the standard deviation of song lengths is less than 31 seconds

Additional info: These notes expand on the original slides by providing full definitions, formulas in LaTeX, and structured examples for each type of hypothesis test. The summary table is inferred for clarity and completeness.