Chapter 8, Problem 8.1P

For each of the following, test for the significance of the difference in sample statistics using the five- step model. (HINT: Remember to solve Formula 8.4 before attempting to solve Formula 8.2. Also, in Formula 8.4, perform the mathematical operations in the proper sequence. First square each sample standard deviation, then divide by the proper $N$, add the resultant values, and then find the square root of the sum.)

a.

Sample 1	Sample 2
${\bar{X}}_1=72.5$	${\bar{X}}_2=76.0$
$s_1=14.3$	$s_2=10.2$
$N_1=136$	$N_2=257$

b.

Sample 1	Sample 2
${\bar{X}}_1=107$	${\bar{X}}_2=103$
$s_1=14$	$s_2=17$
$N_1=175$	$N_2=200$

To determine

(a)

To find:

The significant difference in the sample statistics for the two samples.

Answer to Problem 8.1P

Solution:

There is a significant difference between the sample statistics of two samples.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1	Sample 2
${\overline{X}}_1=72.5$	${\overline{X}}_2=76.0$
$s_1=14.3$	$s_2=10.2$
$N_1=136$	$N_2=257$

The five step model for hypothesis testing:

Step 1. Making assumptions and meeting test requirements.

Step 2. Stating the null hypothesis.

Step 3. Selecting the sampling distribution and establishing the critical region.

Step 4. Computing test statistics.

Step 5. Making a decision and interpreting the results of the test.

Formula used:

The formula to calculate the sampling distribution of the differences in sample means is given by,

$Z\left(\text{obtained}\right)=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{{\sigma }_{\overline{X}-\overline{X}}}$

Where, ${\overline{X}}_1$ and ${\overline{X}}_2$ is the mean of first and second sample respectively,

${\mu }_1$ and ${\mu }_2$ is the mean of first and second population respectively,

${\sigma }_{\overline{X}-\overline{X}}$ is the standard deviation and the formula to calculate ${\sigma }_{\overline{X}-\overline{X}}$ is given by,

${\sigma }_{\overline{X}-\overline{X}}=\sqrt{\frac{s^2{}_1}{N_1-1}+\frac{s^2{}_2}{N_2-1}}$

Where, $N_1$ and $N_2$ is the number of first and second population respectively.

Calculation:

From the given information, the sample size of the first sample is 136, the sample size of the second sample is 257, the sample mean of the first sample is 72.5, the sample mean of the second sample is 76.0, the sample standard deviation of the first sample is 14.3, and the sample standard deviation of the second sample is 10.2.

As the significant difference in the sample statistics is to be determined, a two tailed test is applied.

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Consider independent random samples.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. State the null hypothesis.

The statement of the null hypothesis is that there is no significant difference in the sample s of the population. Thus, the null and the alternative hypotheses are,

$H_0:{\mu }_1={\mu }_2$

$H_1:{\mu }_1\ne {\mu }_2$

Step 3. Selecting the sampling distribution and establishing the critical region.

Since, the sample size is large, Z distribution can be used.

Thus, the sampling distribution is Z distribution.

The level of significance is,

$\alpha =0.05$

Area of critical region is,

$Z(\text{critical})=\pm 1.96$

Step 4. Compute test statistics.

The population standard deviations are unknown.

The formula to calculate ${\sigma }_{\overline{X}-\overline{X}}$ is given by,

${\sigma }_{\overline{X}-\overline{X}}=\sqrt{\frac{s^2{}_1}{N_1-1}+\frac{s^2{}_2}{N_2-1}}$

Substitute 14.3 for $s_1$, 10.2 for $s_2$, 136 for $N_1$, and 257 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& \sqrt{\frac{{\left(14.3\right)}^2}{136-1}+\frac{{\left(10.2\right)}^2}{257-1}}\\ & =& \sqrt{\frac{204.49}{135}+\frac{104.04}{256}}\\ & =& \sqrt{1.5147+0.4064}\\ & =& \sqrt{1.9211}\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& 1.3861\\ & \approx & 1.39\to \left(1\right)\end{array}$

The sampling distribution of the differences in sample means is given by,

$Z\left(\text{obtained}\right)=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{{\sigma }_{\overline{X}-\overline{X}}}$

Under the null hypotheses,

${\mu }_1-{\mu }_2=0$

Substitute 0 for ${\mu }_1-{\mu }_2$ in the above mentioned formula,

$Z\left(\text{obtained}\right)=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)}{{\sigma }_{\overline{X}-\overline{X}}}$

From equation $\left(1\right)$ substitute 72.5 for ${\overline{X}}_1$, 76.0 for ${\overline{X}}_2$, and 1.39 for ${\sigma }_{\overline{X}-\overline{X}}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(72.5-76.0\right)}{1.39}\\ & =& \frac{-3.5}{1.39}\\ & =& -2.52\end{array}$

Thus, the obtained Z value is $-2.52$.

Step 5. Making a decision and interpreting the results of the test.

Compare the test statistic with the critical $Z$ value. The $Z$ score falls in the rejection region. This implies that there is a significant difference between the two samples. The decision to reject the null hypothesis has only 0.05 probability of being incorrect.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples.

To determine

(b)

To find:

The significant difference in the sample statistics of the two samples.

Answer to Problem 8.1P

Solution:

There is a significant difference between the sample statistics of two samples.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1	Sample 2
${\overline{X}}_1=107$	${\overline{X}}_2=103$
$s_1=14$	$s_2=17$
$N_1=175$	$N_2=200$

The five step model for hypothesis testing:

Step 1. Making assumptions and meeting test requirements.

Step 2. Stating the null hypothesis.

Step 3. Selecting the sampling distribution and establishing the critical region.

Step 4. Computing test statistics.

Step 5. Making a decision and interpreting the results of the test.

Formula used:

The formula to calculate the sampling distribution of the differences in sample means is given by,

$Z\left(\text{obtained}\right)=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{{\sigma }_{\overline{X}-\overline{X}}}$

Where, ${\overline{X}}_1$ and ${\overline{X}}_2$ is the mean of first and second sample respectively,

${\mu }_1$ and ${\mu }_2$ is the mean of first and second population respectively,

${\sigma }_{\overline{X}-\overline{X}}$ is the standard deviation and the formula to calculate ${\sigma }_{\overline{X}-\overline{X}}$ is given by,

${\sigma }_{\overline{X}-\overline{X}}=\sqrt{\frac{s^2{}_1}{N_1-1}+\frac{s^2{}_2}{N_2-1}}$

Where, $N_1$ and $N_2$ is the number of first and second population respectively.

Calculation:

From the given information, the sample size of the first sample is 175, the sample size of the second sample is 200, the sample mean of the first sample is 107, the sample mean of the second sample is 103, the sample standard deviation of the first sample is 14, and the sample standard deviation of the second sample is 17.

As the significant difference in the sample statistics is to be determined, a two tailed test is applied.

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Consider independent random samples.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. State the null hypothesis.

The statement of the null hypothesis is that there is no significant difference in the sample s of the population. Thus, the null and the alternative hypotheses are,

$H_0:{\mu }_1={\mu }_2$

$H_1:{\mu }_1\ne {\mu }_2$

Step 3. Selecting the sampling distribution and establishing the critical region.

Since, the sample size is large, Z distribution can be used.

Thus, the sampling distribution is Z distribution.

The level of significance is,

$\alpha =0.05$

Area of critical region is,

$Z(\text{critical})=\pm 1.96$

Step 4. Compute test statistics.

The population standard deviations are unknown.

The formula to calculate ${\sigma }_{\overline{X}-\overline{X}}$ is given by,

${\sigma }_{\overline{X}-\overline{X}}=\sqrt{\frac{s^2{}_1}{N_1-1}+\frac{s^2{}_2}{N_2-1}}$

Substitute 14 for $s_1$, 17 for $s_2$, 175 for $N_1$, and 200 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& \sqrt{\frac{{\left(14\right)}^2}{175-1}+\frac{{\left(17\right)}^2}{200-1}}\\ & =& \sqrt{\frac{196}{174}+\frac{289}{199}}\\ & =& \sqrt{1.1264+1.4523}\\ & =& \sqrt{2.5787}\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& 1.6058\\ & \approx & 1.61\to \left(2\right)\end{array}$

The sampling distribution of the differences in sample means is given by,

$Z\left(\text{obtained}\right)=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{{\sigma }_{\overline{X}-\overline{X}}}$

Under the null hypotheses,

${\mu }_1-{\mu }_2=0$

Substitute 0 for ${\mu }_1-{\mu }_2$ in the above mentioned formula,

$Z\left(\text{obtained}\right)=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)}{{\sigma }_{\overline{X}-\overline{X}}}$

From equation $\left(2\right)$ substitute 107 for ${\overline{X}}_1$, 103 for ${\overline{X}}_2$, and 1.61 for ${\sigma }_{\overline{X}-\overline{X}}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(107-103\right)}{1.61}\\ & =& \frac{4}{1.61}\\ & =& 2.48\end{array}$

Thus, the obtained Z value is $2.48$.

Step 5. Making a decision and interpreting the results of the test.

Compare the test statistic with the critical $Z$ value. The $Z$ score falls in the rejection region. This implies that there is a significant difference between the two samples. The decision to reject the null hypothesis has only 0.05 probability of being incorrect.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples.