Exercise 9-13 Algo
Consider the following hypotheses:
H_{0}: μ ≤ 73.0
H_{A}: μ > 73.0
A sample of 51 observations yields a sample mean of 74.3. Assume
that the sample is drawn from a normal population with a population
standard deviation of 3.7. (You may find it useful to
reference the appropriate table: z table
or t table)
a-1. Find the p-value.
p-value < 0.01
a-2. What is the conclusion if α =
0.01?
Do not reject H_{0} since the p-value is smaller than α.
Do not reject H_{0} since the p-value is greater than α.
Reject H_{0} since the p-value is smaller than α.
Reject H_{0} since the p-value is greater than α.
a-3. Interpret the results at α =
0.01.
We cannot conclude that the sample mean is greater than 73.
We conclude that the sample mean is greater than 73.
We cannot conclude that the population mean is greater than 73.
We conclude that the population mean is greater than 73.
b-1. Calculate the p-value if the above
sample mean was based on a sample of 118 observations.
p-value < 0.01
b-2. What is the conclusion if α =
0.01?
Reject H_{0} since the p-value is smaller than α.
Reject H_{0} since the p-value is greater than α.
Do not reject H_{0} since the p-value is smaller than α.
Do not reject H_{0} since the p-value is greater than α.
b-3. Interpret the results at α =
0.01.
We conclude that the sample mean is greater than 73.
We cannot conclude that the sample mean is greater than 73.
We conclude that the population mean is greater than 73.
We cannot conclude that the population mean is greater than 73.
a-1) Zstat = (xbar - mu)/(s/√n).
Therefore, Zstat becomes (74.3-73)/(3.7/√51) = 2.5 . Since it's an upper tail test, therefore p-value is given by (1- ¢(z)) = (1- ¢(2.5)) = 1-0.9938 = 0.0062.
a-2) At alpha = 0.01, we reject the null hypothesis since p<alpha. (Option A)
a-3) We conclude that the population mean is greater than 73 since we have rejected the null. (Option D)
b-1) For n = 118, Zstat becomes (74.3-73)/(3.7/√118) = 3.82 and thereby p-value becomes (1- ¢(3.82)) = (1- 1) = 0
b-2) We'll reject null hypothesis since again p-value is less than alpha (Option A)
b-3) We again conclude that the population mean is greater than 73 (Option C)
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