Hypothesis Testing in Statistics

Learning Objectives

• LO 9.1: Define the null and alternative hypotheses.
• LO 9.2: Differentiate between Type I and Type II errors.
• LO 9.3: Perform a hypothesis test for a population mean when σ is known.
• LO 9.4: Perform a hypothesis test for a population mean when σ is unknown.
• LO 9.5: Perform a hypothesis test for a population proportion.

Introductory Case: Undergraduate Study Habits

• A recent study indicates a decrease in the average number of hours college students study per week over the last five decades.
• In 1961, students studied an average of 24 hours per week.
• Current students study an average of 14 hours per week.
• Aaliyah Knight wants to determine if the study trend reflects the students at her university.
• Susan randomly selected 35 students to ask about their average study time per week.
• Aaliyah aims to determine if the mean study time is below the 1961 average of 24 hours/week.
• She also wants to determine if the mean study time differs from today's national average of 14 hours/week.

Introduction to Hypothesis Testing

• Hypothesis testing resolves conflicts between two competing hypotheses.
• The null hypothesis is the presumed default state, denoted as H0.
• The alternative hypothesis contradicts the null hypothesis and is denoted as HA.
• Hypothesis testing determines if sample evidence contradicts the null hypothesis.
• The ultimate goal is to determine if the null hypothesis can be rejected.
• A test about μ is based on X.
• E(X) = μ.
• se(X) = σ / √n.
• It is essential that the sampling distribution of X is approximately normal (underlying population is normal or n ≥ 30).
• μ0 represents the hypothesized value of the population mean.
• The test statistic accounts for variability and is evaluated at μ = μ0, z = (X - μ0) / (σ / √n).
• The test statistic is used to find the p-value.
• Every hypothesis confronts evidence that either substantiates or refutes it.
• Determining the validity of an assumption is hypothesis testing.
• Hypothesis testing is the second type of inferences about population parameters.

Decisions and Errors in Hypothesis Testing

• Two decisions can be made: rejecting the null hypothesis or not rejecting it.
• The null hypothesis is rejected when sample evidence is inconsistent with that hypothesis.
• Rejecting the null hypothesis is like finding someone innocent in criminal court, due to lack of evidence.
• Not rejecting the null hypothesis means the sample evidence is not inconsistent with it.
• Sample information might not be inconsistent with the null, which doesn't prove it's true.
• In the criminal court analogy, this is akin to finding someone guilty with enough evidence to convict.
• Type I error involves rejecting the null hypothesis when it is actually true.
• Type II error involves not rejecting the null hypothesis when it is false.
• Rejecting a true null hypothesis is a Type I error.
• Not rejecting a false null hypothesis is a Type II error.
• It's not always straightforward to determine which error type carries more severe consequences.
• Given fixed evidence, there's a trade-off between these errors.
• Reducing the chance of a Type I error often raises the chance of a Type II error, and vice versa.
• Increasing the sample size can lower both types of errors.
• A Type I error is denoted by α, while a Type II error is denoted by β.
• The only way to lower both α and β is by increasing n.
• We can reduce α only at the expense of a higher β and vice versa.

Formulating Hypotheses

• Formulating the two competing hypotheses is crucial in hypothesis testing.
• The test's conclusion hinges on how these hypotheses are stated.
• One-tailed tests use < or > in the alternative hypothesis.
• Two-tailed tests use ≠ in the alternative hypothesis.

Null Hypothesis

• The null hypothesis represents the "status quo" or "business as usual."
• It is specified with ≤, =, or ≥.

Alternative Hypothesis

• The alternative hypothesis contests the status quo, seeking to establish something new.
• It is specified with the opposite of what is in the null: ≠, >, or <.
• The relative cost of Type I and Type II errors determines the optimal choice of α and β.
• Determining these costs can be challenging.
• Typically, the management of a firm decides the optimal level of Type I and Type II errors.
• A statistician's responsibility is to conduct the hypothesis test for a chosen value of α.
• General steps to formulating hypotheses include:
  • Identifying the parameter of interest.
  • Determining if the test is one- or two-tailed.
  • Including an equality sign in the null hypothesis and using the alternative to establish a claim.

Hypothesis Testing Examples

• A trade group predicts back-to-school spending will average $606.40 per family.
  • If this prediction is wrong, a new economic model is needed.
  • The parameter of interest is μ, the average back-to-school spending.
  • This is a two-tailed test because the goal is to determine if the population mean differs from $606.40 (≠).
  • H0: μ = 606.40 and HA: μ ≠ 606.40.
• A research analyst aims to test the claim that over 50% of households will tune in for a TV episode.
  • The parameter of interest is p, the proportion of households.
  • The test is one-tailed since the analyst wants to determine if p > 0.50.
  • H0: p ≤ 0.50 and HA: p > 0.50.

Hypothesis Test for Population Mean (Sigma Known)

• Basic methodology for hypothesis testing in this context.
• It is true that  is rarely known.
• Instances where  is stable and can be determined from prior experience.
• In these cases, treat  as known.
• This does not change the general procedure or principles.
• Assume the null hypothesis is true initially.
  • Then, you determine if the sample evidence contradicts this assumption.
  • This approach is comparable to the principle in criminal court: the individual is innocent until proven guilty.

Approaches

• P-value: reported by statistical software, used by researchers.
• Critical value: computer is not available, overview in appendix.

P-Value Interpretation

• The P-value is the likelihood of observing a sample mean as extreme as or more extreme than the one calculated from the sample, assuming the null hypothesis is true (μ = μ0).
• The probability of making a Type I error is defined as α.
  • 100α% represents the significance level.
  • The value of α needs to be chosen before implementing a test
  • Common values for α include 0.01, 0.05, and 0.10.
• The P-value stands for the observed probability of committing a Type I error.
• Decision-Making with P-Value:
  • Reject the null hypothesis if the P-value is less than α.
  • Do not reject the null hypothesis if the P-value is greater than or equal to α.

P-Value Calculation Based on Alternative Hypothesis

• The P-value's exact form depends on how the alternative hypothesis is specified.
• μ0 stands for the hypothesized value of the population mean.
• The test statistic, z = (x - μ0)/(σ/√n), is used to calculate the P-value.
• If HA: μ > μ0, the P-value is the right-tail probability: P(Z ≥ z ).
• If HA: μ < μ0, the P-value is the left-tail probability: P(Z ≤ z ).
• If HA: μ ≠ μ0, the P-value is the two-tail probability.
  • Specifically, it is 2P ( Z ≥ z ) if z > 0 or 2 P ( Z ≤ z ) if z < 0.

Four-Step Procedure for Hypothesis Testing

• Specify the Null and Alternative Hypotheses:
• State the null hypothesis (H0) and the alternative hypothesis (HA). Specify the Significance Level:
• Determine the acceptable level of Type I error (α). Calculate the Test Statistic and P-value:
• Compute the appropriate test statistic and determine its associated P-value. State the Conclusion and Interpret the Results:
• Based on the P-value and significance level, decide whether to reject the null hypothesis.
• Provide a clear interpretation of the results in the context of the problem. Four-step procedure approach:
• Specify the null and the alternative hypotheses.
• Specify the significance level. Calculate the test statistic and the p-value.
• State the conclusion and interpret the results.
  • Communicate results clearly, interpreting them for the claim regarding the population parameter.

Hypothesis Testing Example: Back-to-School Spending

• A sample of 30 households showed that the sample mean of back-to-school spending is $622.85.
• It is believed that back-to-school spending is normally distributed with a population standard deviation of $65.
• An analyst wishes to test if the average back-to-school spending differs from $606.40 per family predicted by the trade group at the 5% significance level.

Step 1: Specify Hypotheses

• Null hypothesis (H₀): µ = 606.40, Alternative hypothesis (Hλ): µ ≠ 606.40.
• Step 2: Set Significance Level: α = 0.05. Step 3: Calculate Test Statistic and P-value: z = (622.85 - 606.40) / (65 / √30) = 1.3862
• P-value for two-tailed test: 2 * P(Z ≥ 1.3862). P(Z ≥ 1.3862) = 1 - 0.9177 = 0.0823.
• P-value = 2 * 0.0823 = 0.1647.
• Step 4: State Conclusion: Since 0.1647 > 0.05, do not reject the null hypothesis.
• Conclusion: At the 5% significance level, it cannot be concluded that average back-to-school spending differs from $606.40 per family. The sample data does not support the research analysts claims.

Confidence Intervals in Hypothesis Testing

• Confidence intervals can be used for conducting two-tailed hypothesis tests.
• Recall the confidence interval for μ : x ± zα /2 √(σ / n) or[ x - zα /2 √(σ / n), x + zα /2 √(σ / n)]
• For HA: μ ≠ μ0: Reject H0 if μ0 does not fall within the confidence interval. Do not reject H0 if μ0 falls within the confidence interval.

Using Confidence Intervals: Back-to-School Spending Example

• For back-to-school spending, where the sample mean for 30 households is $622.85.
• Assume back-to-school spending is normally distributed with σ = $65.
• Test if average spending differs from the trade group's prediction of $606.40 at α = 0.05. Specify Hypotheses and Level of Significance: H0: μ = $606.40, HA: μ ≠ $606.40, α = 0.05.
• Calculate the Confidence Interval: The confidence interval is x ± zα / 2 √(σ / n) = 622.85 ± 1.96 (65 / √30) = 622.85 ± 23.26.
• Find Resulting interval: [599.59, 646.11].
• Conclusion: Since hypothesized mean μ0 = $606.40 falls within the 95% confidence interval, we do not reject H0. We arrive at the same conclusion as with the p-value approach: average back-to-school spending differs from $606.40 per family isn't supported.