Essentials Of Statistics
Essentials Of Statistics
4th Edition
ISBN: 9781305093836
Author: HEALEY, Joseph F.
Publisher: Cengage Learning,
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Chapter 8, Problem 8.15P
To determine

(a)

To find:

The significant difference between two sample proportions.

Expert Solution
Check Mark

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)
Ps1=0.37 Ps2=0.31
N1=202 N2=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(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Where, Ps1 and Ps2 is the proportion of first and second sample respectively,

Pu1 and Pu2 is the proportion of first and second population respectively,

σpp is the standard deviation and the formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

Where, N1 and N2 is the number of first and second population respectively.

And Ps1 is the population proportion and the formula to Ps1 is given by,

Pu=N1Ps1+N2Ps2N1+N2

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,

H0:Pu1=Pu2

H1:Pu1Pu2

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,

α=0.05

Area of critical region is,

Z(critical)=±1.96

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate Pu is given by,

Pu=N1Ps1+N2Ps2N1+N2

Substitute 0.37 for Ps1, 0.31 for Ps2, 202 for N1, and 246 for N2 in the above mentioned formula,

Pu=202×0.37+246×0.31202+246=74.74+76.26448=151448=0.34......(1)

The formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

From equation (1), substitute 0.34 for Pu, 202 for N1, and 246 for N2 in the above mentioned formula,

σpp=0.34(10.34)202+246202×246=0.34×0.6644849692=0.22440.009=0.4737×0.0949

Simplify further,

σpp=0.04490.04.......(2)

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

Z(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Under null hypothesis, (Pu1Pu2)=0

Substitute 0 for (Pu1Pu2) in the above mentioned formula,

Z(obtained)=(Ps1Ps2)σpp

From equation (2) substitute 0.37 for Ps1, 0.31 for Ps2, and 0.04 for σpp in the above mentioned formula,

Z(obtained)=(0.370.31)0.04=0.060.04=1.5

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.

Expert Solution
Check Mark

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)
Ps1=0.47 Ps2=0.58
N1=251 N2=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(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Where, Ps1 and Ps2 is the proportion of first and second sample respectively,

Pu1 and Pu2 is the proportion of first and second population respectively,

σpp is the standard deviation and the formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

Where, N1 and N2 is the number of first and second population respectively.

And Ps1 is the population proportion and the formula to Ps1 is given by,

Pu=N1Ps1+N2Ps2N1+N2

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,

H0:Pu1=Pu2

H1:Pu1<Pu2

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,

α=0.05

Area of critical region is,

Z(critical)=1.65

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate Pu is given by,

Pu=N1Ps1+N2Ps2N1+N2

Substitute 0.47 for Ps1, 0.58 for Ps2, 251 for N1, and 351 for N2 in the above mentioned formula,

Pu=251×0.47+351×0.58251+351=117.97+203.58602=321.55602=0.53......(3)

The formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

From equation (3), substitute 0.53 for Pu, 251 for N1, and 351 for N2 in the above mentioned formula,

σpp=0.53(10.53)251+351251×351=0.53×0.4760288101=0.24910.0068=0.4991×0.0825

Simplify further,

σpp=0.04120.04...........(4)

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

Z(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Under null hypothesis, (Pu1Pu2)=0

Substitute 0 for (Pu1Pu2) in the above mentioned formula,

Z(obtained)=(Ps1Ps2)σpp

From equation (4) substitute 0.47 for Ps1, 0.58 for Ps2, and 0.04 for σpp in the above mentioned formula,

Z(obtained)=(0.470.58)0.04=0.110.04=2.75

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.

Expert Solution
Check Mark

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)
Ps1=0.45 Ps2=0.53
N1=399 N2=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(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Where, Ps1 and Ps2 is the proportion of first and second sample respectively,

Pu1 and Pu2 is the proportion of first and second population respectively,

σpp is the standard deviation and the formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

Where, N1 and N2 is the number of first and second population respectively.

And Ps1 is the population proportion and the formula to Ps1 is given by,

Pu=N1Ps1+N2Ps2N1+N2

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,

H0:Pu1=Pu2

H1:Pu1Pu2

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,

α=0.05

Area of critical region is,

Z(critical)=±1.96

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate Pu is given by,

Pu=N1Ps1+N2Ps2N1+N2

Substitute 0.45 for Ps1, 0.53 for Ps2, 399 for N1, and 509 for N2 in the above mentioned formula,

Pu=399×0.45+509×0.53399+509=179.55+269.77908=449.32908=0.49........(5)

The formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

From equation (5), substitute 0.49 for Pu, 399 for N1, and 509 for N2 in the above mentioned formula,

σpp=0.49(10.49)399+509399×509=0.49×0.51908203091=0.24990.0045=0.4999×0.0671

Simplify further,

σpp=0.03350.03......(6)

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

Z(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Under null hypothesis, (Pu1Pu2)=0

Substitute 0 for (Pu1Pu2) in the above mentioned formula,

Z(obtained)=(Ps1Ps2)σpp

From equation (6) substitute 0.45 for Ps1, 0.53 for Ps2, and 0.03 for σpp in the above mentioned formula,

Z(obtained)=(0.450.53)0.03=0.080.03=2.67

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.

Expert Solution
Check Mark

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)
X¯1=4.18 X¯2=3.38
s1=7.21 s2=5.92
N1=431 N2=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(obtained)=(X¯1X¯2)(μ1μ2)σX¯X¯

Where, X¯1 and X¯2 is the mean of first and second sample respectively,

μ1 and μ2 is the mean of first and second population respectively,

σX¯X¯ is the standard deviation and the formula to calculate σX¯X¯ is given by,

σX¯X¯=s21N11+s22N21

Where, N1 and N2 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,

H0:μ1=μ2

H1:μ1>μ2

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,

α=0.05

Area of critical region is,

Z(critical)=1.65

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate σX¯X¯ is given by,

σX¯X¯=s21N11+s22N21

Substitute 7.21 for s1, 5.92 for s2, 431 for N1, and 535 for N2 in the above mentioned formula,

σX¯X¯=(7.21)24311+(5.92)25351=51.9841430+35.0464534=0.1209+0.0656=0.1865

Simplify further,

σX¯X¯=0.43190.43...(7)

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

Z(obtained)=(X¯1X¯2)(μ1μ2)σX¯X¯

Under the null hypotheses,

μ1μ2=0

Substitute μ1μ2=0 in the above mentioned formula,

Z(obtained)=(X¯1X¯2)σX¯X¯

From equation (7) substitute 4.18 for X¯1, 3.38 for X¯2, and 0.43 for σX¯X¯ in the above mentioned formula,

Z(obtained)=(4.183.38)0.43=0.80.43=1.86

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.

Expert Solution
Check Mark

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)
X¯1=3.19 X¯2=3.99
s1=2.60 s2=2.72
N1=641 N2=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(obtained)=(X¯1X¯2)(μ1μ2)σX¯X¯

Where, X¯1 and X¯2 is the mean of first and second sample respectively,

μ1 and μ2 is the mean of first and second population respectively,

σX¯X¯ is the standard deviation and the formula to calculate σX¯X¯ is given by,

σX¯X¯=s21N11+s22N21

Where, N1 and N2 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,

H0:μ1=μ2

H1:μ1<μ2

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,

α=0.05

Area of critical region is,

Z(critical)=1.65

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate σX¯X¯ is given by,

σX¯X¯=s21N11+s22N21

Substitute 2.60 for s1, 2.72 for s2, 641 for N1, and 808 for N2 in the above mentioned formula,

σX¯X¯=(2.60)26411+(2.72)28081=6.76640+7.3984807=0.0106+0.0092=0.0198

Simplify further,

σX¯X¯=0.14070.14...(8)

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

Z(obtained)=(X¯1X¯2)(μ1μ2)σX¯X¯

Under the null hypotheses,

μ1μ2=0

Substitute μ1μ2=0 in the above mentioned formula,

Z(obtained)=(X¯1X¯2)σX¯X¯

From equation (8) substitute 3.19 for X¯1, 3.99 for X¯2, and 0.14 for σX¯X¯ in the above mentioned formula,

Z(obtained)=(3.193.99)0.14=0.80.14=5.71

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.

Expert Solution
Check Mark

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)
X¯1=1.49 X¯2=1.93
s1=1.50 s2=1.50
N1=635 N2=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(obtained)=(X¯1X¯2)(μ1μ2)σX¯X¯

Where, X¯1 and X¯2 is the mean of first and second sample respectively,

μ1 and μ2 is the mean of first and second population respectively,

σX¯X¯ is the standard deviation and the formula to calculate σX¯X¯ is given by,

σX¯X¯=s21N11+s22N21

Where, N1 and N2 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,

H0:μ1=μ2

H1:μ1<μ2

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,

α=0.05

Area of critical region is,

Z(critical)=1.65

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate σX¯X¯ is given by,

σX¯X¯=s21N11+s22N21

Substitute 1.50 for s1, 1.50 for s2, 635 for N1, and 803 for N2 in the above mentioned formula,

σX¯X¯=(1.50)26351+(1.50)28031=2.25634+2.25802=0.0035+0.0028=0.0063

Simplify further,

σX¯X¯=0.07940.08....(9)

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

Z(obtained)=(X¯1X¯2)(μ1μ2)σX¯X¯

Under the null hypotheses,

μ1μ2=0

Substitute μ1μ2=0 in the above mentioned formula,

Z(obtained)=(X¯1X¯2)σX¯X¯

From equation (9) substitute 1.49 for X¯1, 1.93 for X¯2, and 0.08 for σX¯X¯ in the above mentioned formula,

Z(obtained)=(1.491.93)0.08=0.440.08=5.5

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.

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