Question

Consider the test \({H_0}:\mu = 70\) versus \({H_a}:\mu \ne 70\) using a large sample of size n = 400. Assume\(\sigma = 20\).

a. Describe the sampling distribution of\(\bar x\).

b. Find the value of the test statistic if\(\bar x = 72.5\).

c. Refer to part b. Find the p-value of the test.

d. Find the rejection region of the test for\(\alpha = 0.01\).

e. Refer to parts c and d. Use the p-value approach to

make the appropriate conclusion.

f. Repeat part e, but use the rejection region approach.

g. Do the conclusions, parts e and f, agree?

Short answer

a. Since the sample size is sufficiently large and the population standard deviation is known, the sampling distribution of the sample mean will be normal.

b. The test statistic is\(z = 2.5\).

c. The p-value of the test is 0.0062.

d. The rejection region is\(z > 2.33\).

e. Reject the claim that the population mean is 70.

f. Reject the claim that the true mean is 70.

g. The conclusion is the same by using the two approaches, the p-value, and the rejection method.

Step-by-step solution

Given information

A hypothesis testing problem is:\({H_0}:\mu = 70\)versus\({H_a}:\mu \ne 70\).

The population standard deviation is 20.

A random sample of size 400 is selected.

The sample mean is \(\bar x = 72.5\).

Describing the sampling distribution of the sample mean

a.

Since the sample size is sufficiently large and the population standard deviation is known, the sampling distribution of the sample mean will be normal.

Computing the value of the test statistic

b.

The test statistic is:

\(\begin{aligned}z &= \frac{{\bar x - \mu }}{{\frac{\sigma }{{\sqrt n }}}}\\ &= \frac{{72.5 - 70}}{{\frac{{20}}{{\sqrt {400} }}}}\\ &= 2.5\end{aligned}\).

Therefore, the test statistic is \(z = 2.5\).

Computing the p-value

c.

The p-value for the right-tailed test is:

\(\begin{aligned}p &= P\left( {Z > 2.50} \right)\\ &= 1 - P\left( {Z \le 2.50} \right)\\ &= 1 - 0.9938\\ &= 0.0062\end{aligned}\).

The z-table is used to obtain the probability of a z-score less than or equal to 2.50.

Therefore, the p-value is 0.0062.

Finding the rejection region

d.

The z-critical value at the significance level\(\alpha = 0.01\)for the right-tailed test is:

\({z_{0.01}} = 2.33\).

Here absolute value is taken from the z-table.

Therefore, the rejection region is \(z > 2.33\).

Drawing the conclusion based on the p-value

e.

The p-value for the hypothesis test is 0.0062, which is less than\(\alpha = 0.01\).

Therefore, reject the claim that the population mean is 70 in favor of the mean greater than 70.

Drawing the conclusion based on the rejection region

f.

The rejection region for the hypothesis test is greater than 2.33. Since the value of the test statistic is greater than 2.33, reject the null hypothesis.

Therefore, reject the claim that the true mean is 70.

Drawing the conclusion based on the p-value

g.

The conclusion is the same by using the two approaches p-value and rejection method. Traditionally people used the rejection method, but now software uses the p-value approach to decide the hypothesis test.