Chapter 8: Hypothesis Testing Study Guide

Introduction to Hypothesis Testing and Basics (Section 8-1)

• Hypothesis Definition: In the field of statistics, a hypothesis is defined as a claim or a statement regarding a specific property of a population.

• Hypothesis Test (Test of Significance): This is a formal procedure used for testing a claim about a property of a population. It involves comparing two opposing statements and determining which one is more likely to be true based on available sample information.

• Conducting a Hypothesis Test: The process begins with a claim about a parameter, followed by using sample data to test that claim, and ends with a formal conclusion. The core steps are:

  • Identify the original claim and formulate the hypotheses.

  • Use sample information to perform the test.

  • Formulate a conclusion and write a formal statement regarding the original claim.

The Null and Alternative Hypotheses

• Null Hypothesis ($H_0$): The null hypothesis is a statement asserting that the value of a population parameter (such as a mean $\mu$ or a proportion $p$) is equal to a specific claimed value.

  • Notation Example: $H_0: \mu = 100$ (Read as "$H$-naught is mu equals 100").

• Alternative Hypothesis ($H_1$): The alternative hypothesis is a statement asserting that the parameter has a value that differs from the value stated in the null hypothesis in some specific way.

  • Notation Examples:

    • $H_1: \mu < 100$

    • $H_1: \mu \neq 100$

    • $H_1: \mu > 100$

• Claims and Variable Usage: A claim is the statement being tested. It will eventually be assigned as either the null hypothesis or the alternative hypothesis.

  • If a claim concerns a mean, the variable $\mu$ is used in the hypotheses.

  • If a claim concerns a proportion, the variable $p$ is used in the hypotheses.

Hypothesis Formulation Examples

• Example 1: A professor claims that the proportion of all students who complete a test review is less than $65\%$.

  • $H_0: p = 0.65$

  • $H_1: p < 0.65$

• Example 2: A politician claims that most people support lower taxes. ("Most" implies a proportion greater than $0.5$).

  • $H_0: p = 0.5$

  • $H_1: p > 0.5$

• Example 3: A principal wants to determine if the mean SAT score among high school juniors is $650$.

  • $H_0: \mu = 650$

  • $H_1: \mu \neq 650$

Types of Hypothesis Tests and Test Statistics

• Directionality of Tests: The nature of the test is determined by the symbol used in the alternative hypothesis ($H_1$):

  • Left-tail test: Used when $H_1$ contains the less-than symbol ($H_1: \mu < 100$).

  • Right-tail test: Used when $H_1$ contains the greater-than symbol ($H_1: \mu > 100$).

  • Two-tail test: Used when $H_1$ contains the not-equal-to symbol ($H_1: \mu \neq 100$).

• Test Statistic: This value is calculated based on sample information and the assumption that the null hypothesis is true. It can be found via specific formulas or using a calculator's STAT – TESTS menu.

• Finding Test Statistic for a Proportion (TI-Calculator):

  1. Press STAT and navigate to the TESTS menu.

  2. Select 5: 1PropZTest….

  3. Input the required values from the problem.

  4. Select Calculate or Draw (Note: Ensure StatPlots are off and Y= is clear if using Draw).

• Example 4 (Proportion Test Statistic): Claim is $p < 0.65$. Sample size $n = 340$; sample proportion $\hat{p} = 0.63$. This information is used to find the test statistic $z$.

• Finding Test Statistic for a Mean (TI-Calculator):

  1. Press STAT and navigate to TESTS.

  2. Select 1: Z-Test… or 2: T-Test… based on data provided.

  3. Input values and select Calculate or Draw.

• Example 5 (Mean Test Statistic): Claim is $\mu = 650$. Sample size $n = 54$; sample mean $\bar{x} = 642$; standard deviation $s = 25$. This information is used to find the test statistic $t$ (or $z$ depending on known parameters).

Methods for Reaching a Conclusion

• Determining the Outcome: Success in a hypothesis test results in either "rejecting the null hypothesis" or "failing to reject the null hypothesis."

• Significance Level ($\alpha$): This is the probability value used as the cutoff for determining significant evidence against the null hypothesis. Graphically, it represents the area in the tail(s) of the distribution.

Critical Value Method

• Critical Region (Rejection Region): The area under the curve representing all test statistic values that lead to the rejection of the null hypothesis.

• Critical Value: The boundary value(s) separating the critical region from the values that do not lead to rejection. These are determined by $\alpha$.

• Calculating Critical Values (at Mean $0$, SD $1$):

  • Left-tail: $-z_{\alpha} = \text{invnorm}(\alpha, 0, 1)$

  • Right-tail: $z_{\alpha} = \text{invnorm}(1 - \alpha, 0, 1)$

  • Two-tail: $\pm z_{\alpha} = \pm \text{invnorm}(\frac{\alpha}{2}, 0, 1)$

• Decision Rule:

  • If the test statistic falls within the critical region, reject $H_0$.

  • If the test statistic falls outside the critical region, fail to reject $H_0$.

• Example 6 (Finding Critical Values):

  • Right-tail, $\alpha = 0.04$

  • Left-tail, $\alpha = 0.01$

  • Two-tail, $\alpha = 0.05$

• Example 7: Right-tailed test, critical value = $1.96$, test statistic = $2.18$. Since $2.18 > 1.96$, the statistic is in the rejection region; therefore, reject $H_0$.

• Example 8: Two-tailed test, critical values = $\pm 1.64$, test statistic = $-1.44$. Since $-1.44$ is between $-1.64$ and $1.64$, do not reject $H_0$.

• Example 9: Claim $p > 0.45$, $z = 1.98$, $\alpha = 0.10$. Find critical value and decide to reject or fail to reject.

• Example 10: Claim $p = 0.45$, $z = 2.26$, $\alpha = 0.01$. Find critical value and decide to reject or fail to reject.

P-Value Method

• P-Value Definition: The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

• Decision Rule:

  • If $P \leq \alpha$, reject the null hypothesis.

  • If $P > \alpha$, fail to reject the null hypothesis.

Wording the Final Conclusion

• The way a conclusion is phrased depends on whether the original claim was the null ($H_0$) or the alternative ($H_1$) hypothesis, and whether $H_0$ was rejected.

• Case 1: Original Claim is the Alternative Hypothesis ($H_1$):

  • If $H_0$ is rejected: "There is sufficient evidence to support the claim that [original claim]."

  • If $H_0$ is not rejected: "There is not sufficient evidence to support the claim that [original claim]."

• Case 2: Original Claim is the Null Hypothesis ($H_0$):

  • If $H_0$ is rejected: "There is sufficient evidence to reject the claim that [original claim]."

  • If $H_0$ is not rejected: "There is not sufficient evidence to reject the claim that [original claim]."

• Example 11: Claim (Alternative): Most people support lower taxes. $\alpha = 0.05$, $P = 0.0478$. Since $0.0478 \leq 0.05$, reject $H_0$. Conclusion: There is sufficient evidence to support the claim.

• Example 12: Claim: Proportion completing review is less than $65\%$. Sample: $35\%$ of $435$ students. Subjective estimate: The result of $35\%$ is significantly lower than $65\%$, so there is sufficient evidence to support the claim.

Statistical Errors

• Type I Error: The mistake of rejecting the null hypothesis when it is actually true.

• Type II Error: The mistake of failing to reject the null hypothesis when it is actually false.

• Example 14 (Identifying Type I Error): Rejecting the null hypothesis that the mean SAT score is $650$ when the mean is actually equal to $650$.

• Example 15 (Identifying Type II Error): Failing to reject the null hypothesis that the mean SAT score is $650$ when the mean is actually different from $650$.

Testing a Claim about a Proportion (Section 8-2)

• Requirements:

  1. The sample observations are obtained via a simple random sample.

  2. Binomial distribution conditions are met (Fixed number of trials, independent trials, two categories of success/failure, constant probability of success).

  3. There are at least $5$ successes and at least $5$ failures.

• Calculator Procedure: Use STAT -> TESTS -> 5: 1PropZTest....

• Example 1: Study of McDonald's drive-through accuracy. $33$ inaccurate orders out of $362$. Significance level $\alpha = 0.05$. Test the claim that the inaccurate rate is equal to $10\%$.

• Example 2: Study of $1228$ medical malpractice lawsuits; $856$ were dropped or dismissed. Significance level $\alpha = 0.01$. Test the claim that most lawsuits are dropped or dismissed (i.e., $p > 0.5$).

Testing a Claim about a Mean (Section 8-3)

• Requirements:

  1. Sample observations are a simple random sample.

  2. The population is normally distributed OR the sample size $n > 30$.

• Choosing Between Z-Test and T-Test:

  • If the population standard deviation ($s$ or $\sigma$) is known: Use the normal distribution (1: Z-Test).

  • If the population standard deviation is NOT known: Use the Student T distribution (2: T-Test).

• Calculator Procedure: Use STAT -> TESTS -> 1: Z-Test... or 2: T-Test....

• Example 1 (Mean with Raw Data): Students estimated one minute. $\alpha = 0.05$. Test claim that $\mu = 60\,\text{seconds}$.

  • Data: $69, 81, 39, 65, 42, 21, 60, 63, 66, 48, 64, 70, 96, 91, 65$.

• Example 2 (Mean with Known $\sigma$): National A1C average is $5.2$ with population standard deviation $0.4$. Sample of $35$ pastry chefs has mean $5.38$. Use $\alpha = 0.05$ to test if chefs have a higher A1C level (right-tail test).

• Example 3 (Mean with Unknown $\sigma$): Claim: Soy candle burns longer than $5\,\text{hours}$. Sample $n = 25$. Sample mean $\bar{x} = 5.4\,\text{hours}$, sample standard deviation $s = 0.3\,\text{hours}$. Use $\alpha = 0.04$. Assume normal distribution.