a) The kind of test that should be used is B. One-Tailed test.
b) The test statistic is 2.19.
c) The P-value is 0.05.
d) A. Yes. There is sufficient evidence to support the claim that people do better than 75 on this exam.
e) The 95% confidence interval for the mean score for students using the new text shows that we are 95% confident that the true mean score for all students using the new text is between 77.4 and 87.2.
a) This should be a One-Tailed test because it is based on a directional (or one-sided) hypothesis.
b) The test statistic can be calculated using the formula for a one-sample t-test:
where:
is the sample mean,
μ is the population mean,
s is the standard deviation of the sample, and
n is the sample size.
Given the scores, we can calculate the sample mean (
) as follows:
=
=82.3
Substituting the values into the formula:
= 2.19
Thus, the test statistic is 2.19.
c) Using a t-distribution table or a statistical software, given a t-statistic of 2.19 with 9 degrees of freedom and a one-tailed test, the P-value would be less than 0.05.
d) Since the P-value is less than the significance level (0.05), we reject the null hypothesis.
Thus, since our calculated P-value is less than 0.05, there is sufficient evidence to support the claim that people do better than 75 on this exam.
(e) The 95% confidence interval for the mean score for students using the new text can be calculated using the formula:
± tₐ/2, n − 1⋅
where:
tₐ/2, n − 1 is the t-value for a 95% confidence level with n-1 degrees of freedom.
The exact value for tₐ/2, n − 1 can be found using a t-distribution table or a statistical software. For a 95% confidence level and 9 degrees of freedom, it’s approximately 2.262.
Substituting the values into the formula, we get the confidence interval:
82.3 ± 2.262⋅
= [77.4, 87.2]
Thus, the 95% confidence interval for the mean score for students using the new text is [77.4, 87.2].
The implication is that we are 95% confident that the true mean score for all students using the new text is between 77.4 and 87.2.
Complete Question:
A study is conducted to determine if a newly designed text book is more helpful to learning the material than the old edition. The mean score on the final exam for a course using the old edition is 75. Ten randomly selected people who used the new text take the final exam. Their scores are shown in the table below.
Person Test Score
A 68
B 78
C 86
D 70
E 73
F 81
G 93
H 96
I 90
J 88
Use a 0.05 significance level to test the claim that people do better with the new edition. Assume the standard deviation is 10.5.
(a) What kind of test should be used?
A. Two-Tailed
B. One-Tailed (correct)
C. It does not matter.
(b) The test statistic is _______?(rounded to 2 decimals).
(c) The P-value is_____?
(d) Is there sufficient evidence to support the claim that people do better than 75 on this exam?
A. Yes (Correct)
B. No
(e) Construct a 95% confidence interval for the mean score for students using the new text.