Chapter 8.2, Problem 72E

a.

To determine

Compute the P-value.

Check whether the null hypothesis $H_0$ is not rejected at the $\alpha =0.05$ level.

Answer to Problem 72E

The P-value is 0.159.

The null hypothesis $H_0$ is not rejected at the $\alpha =0.05$ level.

Explanation of Solution

Calculation:

The given test hypotheses are:

Null hypothesis: $H_0:\mu =20$.

Alternate hypothesis: $H_1:\mu >20$.

It is given that the sample size $n=100$,sample mean is $\overline{x}=21$ and the assumed standard deviation is $\sigma =10$. It has been stated that the population mean $\mu$ will not be of practical significance unless $\mu >25$.

Assumptions for performing a hypothesis test about $\mu$when $\sigma$is known:

• The samples are taken from the population are simple random samples.
• The sample size should be large that is, $n>30$ or the population should be distributed according to normal.

Requirement check:

• The sample of 100 is assumed to be simple random samples.
• The sample size of 100 is greater than 30. That is, $100>30$.

Denote $\mu$ as the population mean.

It is given that the level of significance $\alpha =0.05$.

Test statistic:

The z-test statistic is:

$z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$,

Where, $\overline{x}$ be the sample mean, $\mu$ be the hypothesized mean, $\sigma$ be the standard deviation and n be the sample size.

Software procedure:

Step by step procedure to find the test statistic using the MINITAB software:

• Choose Stat > Basic Statistics > 1-Sample Z.
• In Summarized data select the Sample size as 100, Sample mean as 21.
• Select Known standard deviation as 10.
• Enter Hypothesized mean as 20.
• Select Options.
• Choose $\text{Mean}>\text{hypothesized mean}$ in Alternate hypothesis.
• Click OK in all dialogue boxes.

The output using Minitab is given below:

From the MINITAB output, the P-value is 0.159.

Thus, the P-value is 0.159.

Decision based on the P-value method:

• If $P-\text{value}\le \alpha$, reject $H_0$.
• If $P-\text{value}>\alpha$, fail to reject $H_0$.

Conclusion:

The significance level is, $\alpha =0.05$ and the P-value is 0.159.

Here, the P-value of 0.159 is greater than the significance level 0.05.

That is, $P\text{-value}\left(=0.159\right)>\alpha \left(=0.05\right)$.

Therefore, the null hypothesis is not rejected.

Hence, it is verified that the null hypothesis is not rejected.

b.

To determine

Compute the P-value.

Check whether the null hypothesis $H_0$ is rejected at the $\alpha =0.05$ level.

Answer to Problem 72E

The P-value is 0.0008.

Yes, the null hypothesis $H_0$ is rejected at the $\alpha =0.05$ level.

Explanation of Solution

Calculation:

It is given that the sample size $n=1,000$,sample mean is $\overline{x}=21$ and the assumed standard deviation $\sigma =10$.

Assumptions for performing a hypothesis test about $\mu$when $\sigma$is known:

• The samples are taken from the population are simple random samples.
• The sample size should be large that is, $n>30$ or the population should be distributed according to normal.

Requirement check:

• The sample of 1,000 is assumed to be simple random samples.
• The sample size of 1,000 is greater than 30. That is, $1,000>30$.

Denote $\mu$ as the population mean.

It is given that the level of significance $\alpha =0.05$.

Test statistic:

Software procedure:

Step by step procedure to find the test statistic using the MINITAB software:

• Choose Stat > Basic Statistics > 1-Sample Z.
• In Summarized data select the Sample size as 1,000, Sample mean as 21.
• Select Known standard deviation as 10.
• Enter Hypothesized mean as 20.
• Select Options.
• Choose $\text{Mean}>\text{hypothesized mean}$ in Alternate hypothesis.
• Click OK in all dialogue boxes.

The output using Minitab is given below:

From the MINITAB output, the P-value is 0.0008.

Thus, the P-value is 0.0008.

Conclusion:

The significance level is, $\alpha =0.05$ and the P-value is 0.0008.

Here, the P-value of 0.0008 is less than the significance level 0.05.

That is, $P\text{-value}\left(=0.0008\right)<\alpha \left(=0.05\right)$.

Therefore, the null hypothesis is rejected.

Hence, it has verified that the null hypothesis is rejected.

c.

To determine

Explain whether the result is likely to be of practical significance.

Answer to Problem 72E

No, the result may not be of practical significance.

Explanation of Solution

Interpretation:

Practical significance:

The practical significance means the real life applicability of a characteristic under study.

It is often observed that even if the tested mean is significant, in common sense or practically, it may not have enough significanceto be considered as practically significant. That is, it suggests that the difference may not be sufficiently large to be considered of practical significance, even if it is statistically significant.

Here, the test of hypothesis gives statistically significant results. The hypothesized mean valueis 20. In 100 sampled individuals,themean is found assignificantlygreater than20.

However, this is not necessarily a great increase for practical purposes. In other words, meangreater than20 is not useful, unless $\mu >25$. There is no evidence to believe the population mean $\mu$ is greater than 25 from this test.

Hence, the result may not be of practical significance.

d.

To determine

Explain the reason the larger sample can be more likely to produce statistically significant results that are not a practical significance.

Explanation of Solution

Increasing in the sample size increases the power of a statistical test. Now, power is the probability of Type II error subtracted from 1. Thus, a larger power suggests smaller probability of Type II error. Again, decreasing the Type II error increases the probability of Type I error, leading to a greater chance of rejecting the null hypothesis when it is true.

As a result, the increase in the sample size increases the chance of rejecting the null hypothesis when it is true. This produces a statistically significant result for larger sample sizes.

Hence, the larger sample can be more likely to produce statistically significant results that are not a practical significance.