Answer to Question #250369 in Statistics and Probability for smilynne

Question

In each problem, provide the following: 

a. State the hypotheses and identify the claim.

b. Find the critical value(s)

c. Find the test value

d. Make the decision

e. Summarize the result

1. A researcher claims that students in a private school have exam scores that are at most 8 points higher than those of students in public schools. Random samples of 60 students from each type of school are selected and given an exam. The results are shown. At ∝= 0.05, test the claim. (Use traditional method)

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Private 𝑿𝟏 = 𝟏𝟏𝟎 𝝈𝟏 = 𝟏𝟓 𝒏𝟏 = 𝟔0

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Public 𝑿𝟐 = 𝟏𝟎𝟐 𝝈𝟐 = 𝟏𝟓 𝒏𝟐 = 𝟔0

Answer 1

$H_0: \mu_1- \mu_2 = 8 \\ H_1: \mu_1- \mu_2 > 8$

Find the critical value for right-tailed test with α=0.05.

The area for right-tailed test is 1-0.05=0.95, which is above the mean.

From the table of “The Standard Normal Distribution”, the critical value for right tail test corresponding to an area of 0.95 is 1.65.

The critical value for right-tailed test with α=0.05 is Zc=1.65.

Test-statistic

$Z = \frac{(\bar{x_1} -\bar{x_2}) - (\mu_1 -\mu_2)}{\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}} \\ Z = \frac{(110-104)-8}{\sqrt{\frac{15^2}{60} + \frac{15^2}{60}}} \\ = \frac{6-8}{\sqrt{3.75+3.75} } \\ = \frac{-2}{2.7386} \\ = -0.73$

Make the decision:

The critical value is 1.65 and the test value is -0.73.

Here, the test value is less than the critical value of Zc. That is, -0.73<1.65.

Therefore, by the rejection rule, do not reject the null hypothesis.

The test value (–0.73) is in the noncritical region as shown in Figure (1).

Summarize the result:

There is not enough evidence to reject the claim that the private school students have exam scores that are at most 8 points higher than those of the students in public school at α=0.05.