Question

Consider the following hypotheses: Ho: μ s 57.1 HA: μ > 57.1 A sample of 22 observations yields a sample mean of 58.2. Assume that the sample is drawn from a normal population with a population standard deviation of 3.6. (You may find it useful to reference the appropriate table: z table or t table) a-1. Find the p-value. pvalue 20.10 Op-value0.01 0.01 s p-value<0.025 0.025 s p-value 0.05 0.05 s p-value<0.10

Answer 1

Solution:

A sample of 22 observations yields a sample mean of 58.2. Assume that the sample is drawn from a normal population with a population standard deviation of 3.6

a-1. Find the p-value.

Answer: $\boldsymbol{0.05\leq p-value\leq 0.10}$

Explanation:

To find the p-value we need to find the test statistic.

The test statistic is:

$\large z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$

$\large =\frac{58.2-57.1}{\frac{3.6}{\sqrt{22}}}$

$\large =1.43$

Now, the p-value is:

$p-value=P(z>1.43)$

Using the standard normal table, we have:

$p-value=P(z>1.43)=0.0764$

Therefore, the  $\boldsymbol{0.05\leq p-value\leq 0.10}$

a-2. What is the conclusion if $\alpha=0.10$ ?

Answer: Reject H₀ since the p-value is smaller than $\alpha$

a-3. Interpret the results at $\alpha=0.10$

Answer: We conclude that the population mean is greater than 57.1

b-1. calculate the p-value if the above sample mean was based on a sample of 108 observations.

Answer: $\boldsymbol{p-value <0.01}$

Explanation:

To find the p-value we need to find the test statistic.

The test statistic is:

$\large z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$

$\large =\frac{58.2-57.1}{\frac{3.6}{\sqrt{108}}}$

$\large =3.18$

Now, the p-value is:

$p-value=P(z>3.18)$

Using the standard normal table, we have:

$p-value=P(z>3.18)=0.0007$

Therefore, the  $\boldsymbol{ p-value<0.01}$

b-2. What is the conclusion if $\alpha=0.10$?

Answer: Reject H₀ since the p-value is smaller than $\alpha$

b-3. Interpret the results at $\alpha=0.10$

Answer: We conclude that the population mean is greater than 57.1