The distribution of the proportion of working females.

Question

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Answer 1

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Section 2:

To determine

To find: The distribution of the proportion of working males.

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.

The Z-score for the value p^M−p^F=0 is calculated as:

Z=(p^M−p^F)−μp^M−p^Fσp^M−p^F=0−0.060.02516=−2.38

The area below the standard normal curve to the left of Z=−2.38 is obtained as 0.0087 from the standard normal table.

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Answer 2

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Section 2:

To determine

To find: The distribution of the proportion of working males.

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.

The Z-score for the value p^M−p^F=0 is calculated as:

Z=(p^M−p^F)−μp^M−p^Fσp^M−p^F=0−0.060.02516=−2.38

The area below the standard normal curve to the left of Z=−2.38 is obtained as 0.0087 from the standard normal table.

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Answer 3

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Section 2:

To determine

To find: The distribution of the proportion of working males.

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.

The Z-score for the value p^M−p^F=0 is calculated as:

Z=(p^M−p^F)−μp^M−p^Fσp^M−p^F=0−0.060.02516=−2.38

The area below the standard normal curve to the left of Z=−2.38 is obtained as 0.0087 from the standard normal table.

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Answer 4

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Section 2:

To determine

To find: The distribution of the proportion of working males.

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.

The Z-score for the value p^M−p^F=0 is calculated as:

Z=(p^M−p^F)−μp^M−p^Fσp^M−p^F=0−0.060.02516=−2.38

The area below the standard normal curve to the left of Z=−2.38 is obtained as 0.0087 from the standard normal table.

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Answer 5

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Section 2:

To determine

To find: The distribution of the proportion of working males.

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.

The Z-score for the value p^M−p^F=0 is calculated as:

Z=(p^M−p^F)−μp^M−p^Fσp^M−p^F=0−0.060.02516=−2.38

The area below the standard normal curve to the left of Z=−2.38 is obtained as 0.0087 from the standard normal table.

Answer 6

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Answer 7

Chapter 5, Problem 87E

(a)

Section 1:

To determine

To find: The distribution of the proportion of working females.

Answer 8

(a)

Section 1:

Expert Solution

Answer to Problem 87E

Solution: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.0192.

Answer 9

(a)

Section 1:

Expert Solution

Answer 10

(a)

Section 1:

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n, where p is the success proportion in the population and n is the size of sample.

Therefore, for the given problem,

μp^F=pF=0.82

And,

σp^F=pF(1−pF)nF=0.82×(1−0.82)400=0.0192

Answer 13

Section 2:

To determine

To find: The distribution of the proportion of working males.

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

Answer 14

Section 2:

To determine

To find: The distribution of the proportion of working males.

Answer 15

Section 2:

Expert Solution

Answer to Problem 87E

Solution: The proportion of males who were engaged in work follows normal distribution with average μp^M=0.88 and standard deviation σp^M=0.01625.

Answer 16

Section 2:

Expert Solution

Answer 17

Section 2:

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Calculation: The sampling distribution of the sample proportion of success p^ is distributed normally with mean μp^=p and standard deviation σp^=p(1−p)n. Therefore, for the given problem,

μp^M=pM=0.88

And,

σp^M=pM(1−pM)nM=0.88×(1−0.88)400=0.01625

Answer 20

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

Answer 21

(b)

To determine

To find: The sampling distribution of the difference of sample proportions.

Answer 22

(b)

Expert Solution

Answer to Problem 87E

Solution: The sampling distribution of (p^M−p^F) is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

Answer 23

(b)

Expert Solution

Answer 24

(b)

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

Calculation: The proportion of females who were engaged in work follows normal distribution with mean μp^F=0.82 and standard deviation σp^F=0.01921.

The proportion of males who were engaged in work follows normal distribution with mean μp^M=0.88 and standard deviation σp^M=0.01625.

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=p^M−p^F and standard deviation σp^M−p^F=σpM2+σpF2. Therefore,

μp^M−p^F=p^M−p^F=0.88−0.82=0.06

And,

σp^M−p^F=σpM2+σpF2=0.019212+0.016252=0.02516

Answer 27

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.

The Z-score for the value p^M−p^F=0 is calculated as:

Z=(p^M−p^F)−μp^M−p^Fσp^M−p^F=0−0.060.02516=−2.38

The area below the standard normal curve to the left of Z=−2.38 is obtained as 0.0087 from the standard normal table.

Answer 28

(c)

To determine

To find: The probability that a higher proportion of females worked than males.

Answer 29

(c)

Expert Solution

Answer to Problem 87E

Solution: The required probability is P(p^F>p^M)=0.0087_.

Answer 30

(c)

Expert Solution

Answer 31

(c)

Expert Solution

Answer 32

Expert Solution

Answer 33

Explanation of Solution

Calculation: The following probability needs to be calculated for the given problem:

P(p^M<p^F)=P(p^M−p^F<0)

The sampling distribution of p^M−p^F is distributed normally with mean μp^M−p^F=0.06 and standard deviation σp^M−p^F=0.02516.

The probability that p^M−p^F is smaller than 0, which is the area to the left of the point 0 under the normal curve. To find the area under the normal curve, first convert the value of sample mean p^M−p^F=0 to a Z-score. The area left to the particular Z-score can be obtained using Table A provided in the book.