Chapter 8, Problem 8.15P

To determine

(a)

To find:

The significant difference between two sample proportions.

Answer to Problem 8.15P

Solution:

There is no significant difference between the sample statistics of two samples proportions and it is concluded that there is no sufficient evidence to conclude that male and females differ in favor of legalization of marijuana.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1 (Males)	Sample 2 (Females)
$P_{s1}=0.37$	$P_{s2}=0.31$
$N_1=202$	$N_2=246$

Approach:

The five step model for hypothesis testing is,

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 proportions of large samples is given by,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)-\left(P_{u1}-P_{u2}\right)}{{\sigma }_{p-p}}$

Where, $P_{s1}$ and $P_{s2}$ is the proportion of first and second sample respectively,

$P_{u1}$ and $P_{u2}$ is the proportion of first and second population respectively,

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

${\sigma }_{p-p}=\sqrt{P_u\left(1-P_u\right)}\sqrt{\frac{N_1+N_2}{N_1{N}_2}}$

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

And $P_{s1}$ is the population proportion and the formula to $P_{s1}$ is given by,

$P_u=\frac{N_1{P}_{s1}+N_2{P}_{s2}}{N_1+N_2}$

Calculation:

As the significant difference in the sample proportions 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:

Independent random samples.

Level of measurement is nominal.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

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

$H_0:P_{u1}=P_{u2}$

$H_1:P_{u1}\ne P_{u2}$

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. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate $P_u$ is given by,

$P_u=\frac{N_1{P}_{s1}+N_2{P}_{s2}}{N_1+N_2}$

Substitute 0.37 for $P_{s1}$, 0.31 for $P_{s2}$, 202 for $N_1$, and 246 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{P}_u& =& \frac{202\times 0.37+246\times 0.31}{202+246}\\ & =& \frac{74.74+76.26}{448}\\ & =& \frac{151}{448}\\ & =& \mathrm{0.34......}(1)\end{array}$

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

${\sigma }_{p-p}=\sqrt{P_u\left(1-P_u\right)}\sqrt{\frac{N_1+N_2}{N_1{N}_2}}$

From equation $\left(1\right)$, substitute 0.34 for $P_u$, 202 for $N_1$, and 246 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{p-p}& =& \sqrt{0.34\left(1-0.34\right)}\sqrt{\frac{202+246}{202\times 246}}\\ & =& \sqrt{0.34\times 0.66}\sqrt{\frac{448}{49692}}\\ & =& \sqrt{0.2244}\sqrt{0.009}\\ & =& 0.4737\times 0.0949\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{p-p}& =& 0.0449\\ & \approx & \mathrm{0.04.......}(2)\end{array}$

The sampling distribution of the differences in sample proportion for large samples is given by,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)-\left(P_{u1}-P_{u2}\right)}{{\sigma }_{p-p}}$

Under null hypothesis, $\left(P_{u1}-P_{u2}\right)=0$

Substitute 0 for $\left(P_{u1}-P_{u2}\right)$ in the above mentioned formula,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)}{{\sigma }_{p-p}}$

From equation $\left(2\right)$ substitute 0.37 for $P_{s1}$, 0.31 for $P_{s2}$, and 0.04 for ${\sigma }_{p-p}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(0.37-0.31\right)}{0.04}\\ & =& \frac{0.06}{0.04}\\ & =& 1.5\end{array}$

Thus, the obtained Z value is $1.5$.

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 acceptance region. This implies that there is no significant difference between the two samples proportions. The decision to accept the null hypothesis has only 0.05 probability of being incorrect. It is concluded that there is no sufficient evidence to conclude that male and females differ in favor of legalization of marijuana.

Conclusion:

Therefore, there is no significant difference between the sample statistics of two samples proportions and it is concluded that there is no sufficient evidence to conclude that male and females differ in favor of legalization of marijuana.

To determine

(b)

To find:

The significant difference between two sample proportions.

Answer to Problem 8.15P

Solution:

There is a significant difference between the sample statistics of two samples proportions and it is concluded that females strongly agree that kids are life’s greatest joy.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1 (Males)	Sample 2 (Females)
$P_{s1}=0.47$	$P_{s2}=0.58$
$N_1=251$	$N_2=351$

Approach:

The five step model for hypothesis testing is,

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 proportions of large samples is given by,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)-\left(P_{u1}-P_{u2}\right)}{{\sigma }_{p-p}}$

Where, $P_{s1}$ and $P_{s2}$ is the proportion of first and second sample respectively,

$P_{u1}$ and $P_{u2}$ is the proportion of first and second population respectively,

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

${\sigma }_{p-p}=\sqrt{P_u\left(1-P_u\right)}\sqrt{\frac{N_1+N_2}{N_1{N}_2}}$

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

And $P_{s1}$ is the population proportion and the formula to $P_{s1}$ is given by,

$P_u=\frac{N_1{P}_{s1}+N_2{P}_{s2}}{N_1+N_2}$

Calculation:

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

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Independent random samples.

Level of measurement is nominal.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

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

$H_0:P_{u1}=P_{u2}$

$H_1:P_{u1}<P_{u2}$

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,

$Z(\text{critical})=-1.65$

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate $P_u$ is given by,

$P_u=\frac{N_1{P}_{s1}+N_2{P}_{s2}}{N_1+N_2}$

Substitute 0.47 for $P_{s1}$, 0.58 for $P_{s2}$, 251 for $N_1$, and 351 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{P}_u& =& \frac{251\times 0.47+351\times 0.58}{251+351}\\ & =& \frac{117.97+203.58}{602}\\ & =& \frac{321.55}{602}\\ & =& \mathrm{0.53......}(3)\end{array}$

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

${\sigma }_{p-p}=\sqrt{P_u\left(1-P_u\right)}\sqrt{\frac{N_1+N_2}{N_1{N}_2}}$

From equation $\left(3\right)$, substitute 0.53 for $P_u$, 251 for $N_1$, and 351 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{p-p}& =& \sqrt{0.53\left(1-0.53\right)}\sqrt{\frac{251+351}{251\times 351}}\\ & =& \sqrt{0.53\times 0.47}\sqrt{\frac{602}{88101}}\\ & =& \sqrt{0.2491}\sqrt{0.0068}\\ & =& 0.4991\times 0.0825\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{p-p}& =& 0.0412\\ & \approx & \mathrm{0.04...........}(4)\end{array}$

The sampling distribution of the differences in sample proportion for large samples is given by,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)-\left(P_{u1}-P_{u2}\right)}{{\sigma }_{p-p}}$

Under null hypothesis, $\left(P_{u1}-P_{u2}\right)=0$

Substitute 0 for $\left(P_{u1}-P_{u2}\right)$ in the above mentioned formula,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)}{{\sigma }_{p-p}}$

From equation $\left(4\right)$ substitute 0.47 for $P_{s1}$, 0.58 for $P_{s2}$, and 0.04 for ${\sigma }_{p-p}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(0.47-0.58\right)}{0.04}\\ & =& \frac{-0.11}{0.04}\\ & =& -2.75\end{array}$

Thus, the obtained Z value is $-2.75$.

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 proportions. The decision to reject the null hypothesis has only 0.05 probability of being incorrect. It is concluded that females strongly agree that kids are life’s greatest joy.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples proportions and it is concluded that females strongly agree that kids are life’s greatest joy.

To determine

(c)

To find:

The significant difference between two sample proportions.

Answer to Problem 8.15P

Solution:

There is a significant difference between the sample statistics of two samples proportions and it is concluded that males and females differ in opinion for voting Obama in 2012.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1 (Males)	Sample 2 (Females)
$P_{s1}=0.45$	$P_{s2}=0.53$
$N_1=399$	$N_2=509$

Approach:

The five step model for hypothesis testing is,

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 proportions of large samples is given by,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)-\left(P_{u1}-P_{u2}\right)}{{\sigma }_{p-p}}$

Where, $P_{s1}$ and $P_{s2}$ is the proportion of first and second sample respectively,

$P_{u1}$ and $P_{u2}$ is the proportion of first and second population respectively,

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

${\sigma }_{p-p}=\sqrt{P_u\left(1-P_u\right)}\sqrt{\frac{N_1+N_2}{N_1{N}_2}}$

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

And $P_{s1}$ is the population proportion and the formula to $P_{s1}$ is given by,

$P_u=\frac{N_1{P}_{s1}+N_2{P}_{s2}}{N_1+N_2}$

Calculation:

As the significant difference in the sample proportions 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:

Independent random samples.

Level of measurement is nominal.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

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

$H_0:P_{u1}=P_{u2}$

$H_1:P_{u1}\ne P_{u2}$

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. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate $P_u$ is given by,

$P_u=\frac{N_1{P}_{s1}+N_2{P}_{s2}}{N_1+N_2}$

Substitute 0.45 for $P_{s1}$, 0.53 for $P_{s2}$, 399 for $N_1$, and 509 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{P}_u& =& \frac{399\times 0.45+509\times 0.53}{399+509}\\ & =& \frac{179.55+269.77}{908}\\ & =& \frac{449.32}{908}\\ & =& \mathrm{0.49........}(5)\end{array}$

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

${\sigma }_{p-p}=\sqrt{P_u\left(1-P_u\right)}\sqrt{\frac{N_1+N_2}{N_1{N}_2}}$

From equation $\left(5\right)$, substitute 0.49 for $P_u$, 399 for $N_1$, and 509 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{p-p}& =& \sqrt{0.49\left(1-0.49\right)}\sqrt{\frac{399+509}{399\times 509}}\\ & =& \sqrt{0.49\times 0.51}\sqrt{\frac{908}{203091}}\\ & =& \sqrt{0.2499}\sqrt{0.0045}\\ & =& 0.4999\times 0.0671\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{p-p}& =& 0.0335\\ & \approx & \mathrm{0.03......}(6)\end{array}$

The sampling distribution of the differences in sample proportion for large samples is given by,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)-\left(P_{u1}-P_{u2}\right)}{{\sigma }_{p-p}}$

Under null hypothesis, $\left(P_{u1}-P_{u2}\right)=0$

Substitute 0 for $\left(P_{u1}-P_{u2}\right)$ in the above mentioned formula,

$Z\left(\text{obtained}\right)=\frac{\left(P_{s1}-P_{s2}\right)}{{\sigma }_{p-p}}$

From equation $\left(6\right)$ substitute 0.45 for $P_{s1}$, 0.53 for $P_{s2}$, and 0.03 for ${\sigma }_{p-p}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(0.45-0.53\right)}{0.03}\\ & =& \frac{-0.08}{0.03}\\ & =& -2.67\end{array}$

Thus, the obtained Z value is $-2.67$.

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 proportions. The decision to reject the null hypothesis has only 0.05 probability of being incorrect. It is concluded that males and females differ in opinion for voting Obama in 2012.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples proportions and it is concluded that males and females differ in opinion for voting Obama in 2012.

To determine

(d)

To find:

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

Answer to Problem 8.15P

Solution:

There is a significant difference between the sample statistics of two samples and it is concluded that males spent more hours at e-mail each week.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1 (Males)	Sample 2 (Females)
${\overline{X}}_1=4.18$	${\overline{X}}_2=3.38$
$s_1=7.21$	$s_2=5.92$
$N_1=431$	$N_2=535$

Approach:

The five step model for hypothesis testing is,

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:

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

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Independent random samples.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. Stating 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,

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

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

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,

$Z(\text{critical})=-1.65$

Step 4. Computing 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 7.21 for $s_1$, 5.92 for $s_2$, 431 for $N_1$, and 535 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& \sqrt{\frac{{\left(7.21\right)}^2}{431-1}+\frac{{\left(5.92\right)}^2}{535-1}}\\ & =& \sqrt{\frac{51.9841}{430}+\frac{35.0464}{534}}\\ & =& \sqrt{0.1209+0.0656}\\ & =& \sqrt{0.1865}\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& 0.4319\\ & \approx & \mathrm{0.43...}(7)\end{array}$

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 ${\mu }_1-{\mu }_2=0$ 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 $\left(7\right)$ substitute 4.18 for ${\overline{X}}_1$, 3.38 for ${\overline{X}}_2$, and 0.43 for ${\sigma }_{\overline{X}-\overline{X}}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(4.18-3.38\right)}{0.43}\\ & =& \frac{0.8}{0.43}\\ & =& 1.86\end{array}$

Thus, the obtained Z value is $1.86$.

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. It is concluded that males spent more hours at e-mail each week.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples and it is concluded that males spent more hours at e-mail each week.

To determine

(e)

To find:

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

Answer to Problem 8.15P

Solution:

There is a significant difference between the sample statistics of two samples and it is concluded that male has less average rate of church attendance compared to females.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1 (Males)	Sample 2 (Females)
${\overline{X}}_1=3.19$	${\overline{X}}_2=3.99$
$s_1=2.60$	$s_2=2.72$
$N_1=641$	$N_2=808$

Approach:

The five step model for hypothesis testing is,

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:

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

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Independent random samples.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. Stating 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,

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

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

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,

$Z(\text{critical})=-1.65$

Step 4. Computing 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 2.60 for $s_1$, 2.72 for $s_2$, 641 for $N_1$, and 808 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& \sqrt{\frac{{\left(2.60\right)}^2}{641-1}+\frac{{\left(2.72\right)}^2}{808-1}}\\ & =& \sqrt{\frac{6.76}{640}+\frac{7.3984}{807}}\\ & =& \sqrt{0.0106+0.0092}\\ & =& \sqrt{0.0198}\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& 0.1407\\ & \approx & \mathrm{0.14...}(8)\end{array}$

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 ${\mu }_1-{\mu }_2=0$ 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 $\left(8\right)$ substitute 3.19 for ${\overline{X}}_1$, 3.99 for ${\overline{X}}_2$, and 0.14 for ${\sigma }_{\overline{X}-\overline{X}}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(3.19-3.99\right)}{0.14}\\ & =& \frac{-0.8}{0.14}\\ & =& -5.71\end{array}$

Thus, the obtained Z value is $-5.71$.

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. It is concluded that male has less average rate of church attendance compared to females.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples and it is concluded that male has less average rate of church attendance compared to females.

To determine

(f)

To find:

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

Answer to Problem 8.15P

Solution:

There is a significant difference between the sample statistics of two samples and it is concluded that males prefer lesser number of children than females.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1 (Males)	Sample 2 (Females)
${\overline{X}}_1=1.49$	${\overline{X}}_2=1.93$
$s_1=1.50$	$s_2=1.50$
$N_1=635$	$N_2=803$

Approach:

The five step model for hypothesis testing is,

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:

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

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Independent random samples.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. Stating 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,

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

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

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,

$Z(\text{critical})=-1.65$

Step 4. Computing 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 1.50 for $s_1$, 1.50 for $s_2$, 635 for $N_1$, and 803 for $N_2$ in the above mentioned formula,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& \sqrt{\frac{{\left(1.50\right)}^2}{635-1}+\frac{{\left(1.50\right)}^2}{803-1}}\\ & =& \sqrt{\frac{2.25}{634}+\frac{2.25}{802}}\\ & =& \sqrt{0.0035+0.0028}\\ & =& \sqrt{0.0063}\end{array}$

Simplify further,

$\begin{array}{rll}{\sigma }_{\overline{X}-\overline{X}}& =& 0.0794\\ & \approx & \mathrm{0.08....}(9)\end{array}$

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 ${\mu }_1-{\mu }_2=0$ 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 $\left(9\right)$ substitute 1.49 for ${\overline{X}}_1$, 1.93 for ${\overline{X}}_2$, and 0.08 for ${\sigma }_{\overline{X}-\overline{X}}$ in the above mentioned formula,

$\begin{array}{rll}Z\left(\text{obtained}\right)& =& \frac{\left(1.49-1.93\right)}{0.08}\\ & =& \frac{-0.44}{0.08}\\ & =& -5.5\end{array}$

Thus, the obtained Z value is $-5.5$.

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. It is concluded that males prefer lesser number of children than females.

Conclusion:

Therefore, there is a significant difference between the sample statistics of two samples and it is concluded that males prefer lesser number of children than females.