7.10 Carrying Out a Test for the Difference of Two Population Means

• Match the hypothesis type with its null and alternative hypotheses:

  Two-tailed ↔️ μ1 = μ2 vs μ1 ≠ μ2

  One-tailed (Left) ↔️ μ1 = μ2 vs μ1 < μ2

  One-tailed (Right) ↔️ μ1 = μ2 vs μ1 > μ2
• What theorem is applicable if the population is not normally distributed but the sample size is large enough?

  Central Limit Theorem
• The pooled variance combines the sample variances to provide a more precise estimate of the common population variance.

  True
• What is the t-statistic if x̄1 = 50, x̄2 = 45, n1 = 20, n2 = 25, and s_p^2 = 22.25?

  3.53
• What does the null hypothesis assume when comparing two population means?

  No difference between means
• Match the hypothesis type with its null and alternative hypotheses:

  Two-tailed ↔️ μ1 = μ2, μ1 ≠ μ2

  One-tailed (Left) ↔️ μ1 = μ2, μ1 < μ2

  One-tailed (Right) ↔️ μ1 = μ2, μ1 > μ2
• Random sampling is a condition for a valid two-sample t-test

  True
• What is the purpose of calculating pooled variance?

  Estimate population variance
• What is the calculated pooled variance if group 1 has n=10 and s^2=25, and group 2 has n=12 and s^2=20?

  22.25
• What is pooled variance used for in statistical analysis?

  Estimating population variance
• What is the first assumption that must be checked for a two-sample t-test?

  Random samples or assignment
• The pooled variance is used to estimate the population variance when comparing two independent groups and assuming their variances are equal
• What does the t-statistic measure in a two-sample t-test?

  Difference between sample means
• When testing if the average math scores of two different schools are equal, the null hypothesis is μ1 = μ2.

  True
• What does the null hypothesis assume when comparing two population means?

  No difference between means
• For the normality assumption, either the populations are normal or the sample sizes are large enough to apply the Central Limit Theorem.


  True
• What is the formula for calculating pooled variance?

  s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}</latex>
• Arrange the variables used in the t-statistic formula with their descriptions:

  1️⃣ x̄1: Mean of group 1

  2️⃣ x̄2: Mean of group 2

  3️⃣ s_p^2: Pooled variance

  4️⃣ n1: Sample size of group 1

  5️⃣ n2: Sample size of group 2
• The t-statistic is approximately 3.53
• In a one-tailed left test, the alternative hypothesis is μ1 > μ2

  False
• To test if the average math scores of one school are higher than another, the alternative hypothesis is μ1 > μ2
• Independence of samples means measurements from one group influence the other

  False
• The pooled variance combines variances from both groups to provide a more precise estimate


  True
• The t-statistic measures the difference between two sample means relative to variability within each sample


  True
• Pooled variance combines sample variances to provide a more precise estimate of the common population variance.


  True
• The degrees of freedom for a two-sample t-test are calculated as: n1 + n2 - 2
• Steps to find the critical t-value in a t-table for a two-tailed test:

  1️⃣ Find the row corresponding to the degrees of freedom.

  2️⃣ Locate the column corresponding to α/2.

  3️⃣ The critical t-value is at the intersection of the row and column.
• In a two-tailed test, the alternative hypothesis is: μ1 ≠ μ2
• One assumption of the two-sample t-test is that the samples must be random
• The Central Limit Theorem allows for reliable estimation of means when sample sizes are large
• The formula for calculating pooled variance is: s_p^2
• The t-statistic measures the difference between two sample means
• The degrees of freedom for a two-sample t-test are calculated using the formula: n1 + n2 - 2
• What two values are needed to find the critical t-value?

  Degrees of freedom and α
• To find the critical t-value, you need to know the degrees of freedom and the alpha level.


  True
• The critical t-value is the value that the t-statistic must exceed to reject the null hypothesis.


  True
• What happens if the t-statistic is less than the critical t-value?

  Fail to reject H0
• If the p-value is less than the significance level, we reject the null hypothesis.
• To state the conclusion of a hypothesis test in context, you must summarize the main findings and relate them to the original research question.
• In a two-tailed test, the alternative hypothesis is μ1 ≠ μ2