An article reported the results of a peer tutoring program to help children learn to read. In the experiment, the children were randomly divided into two groups: the experimental group received peer tutoring along with regular instruction, and the control group received regular instruction with no peer tutoring. There weren₁=n₂= 30children in each group. The Gates-MacGintie Reading Test was given to both groups before instruction began. For the experimental group, the mean score on the vocabulary portion of the test wasx₁=344.5,with sample standard deviations₁=49.7.For the control group, the mean score on the same test wasx₂=353.8,with sample standard deviations₂=50.9.

(a) State the null and alternate hypotheses.

H₀:₁=₂;H₁:₁>₂

H₀:₁=₂;H₁:₁₂

H₀:₁=₂;H₁:₁<₂

H₀:₁₂;H₁:₁=₂

(b) What sampling distribution will you use? What assumptions are you making?

• The standard normal. Both sample sizes are large with known standard deviations.
• The Student'st. Both sample sizes are large with known standard deviations.
• The Student'st. Both sample sizes are large with unknown standard deviations.
• The standard normal. Both sample sizes are large with unknown standard deviations.

(c) What is the value of the sample test statistic? Compute the correspondingzortvalue as appropriate. (Test the difference₁₂.Round your answer to three decimal places.)

(d) Find (or estimate) theP-value.

• P-value > 0.500
• 0.250 <P-value < 0.500
• 0.100 <P-value < 0.250
• 0.050 <P-value < 0.100
• 0.010 <P-value < 0.050
• P-value < 0.010

(e) Sketch the sampling distribution and show the area corresponding to theP-value.

(f) Based on your answers in parts (i)(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

• At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
• At the= 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
• At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
• At the= 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

(g) Interpret your conclusion in the context of the application.

• Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.
• Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.
• Reject the null hypothesis, there is insufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.
• Reject the null hypothesis, there is sufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.

(h) Find a 95% confidence interval for₁₂.

(Round your answers to two decimal places.)

• lower limit=
• upper limit=

(i) Explain the meaning of the confidence interval in the context of the problem.

• Because the interval contains only positive numbers, this indicates that at the 95% confidence level, the population mean score for the experimental group is higher than for the control group.
• Because the interval contains both positive and negative numbers, this indicates that at the 95% confidence level, we cannot say that the population mean scores are different before any instruction began.
• Because the interval contains both positive and negative numbers, this indicates that at the 95% confidence level, the population mean score for the experimental group is higher than for the control group.
• Because the interval contains only negative numbers, this indicates that at the 95% confidence level, the population mean score for the experimental group is lower than for the control group.