Essentials of Business Analytics (MindTap Course List)
Essentials of Business Analytics (MindTap Course List)
2nd Edition
ISBN: 9781305627734
Author: Jeffrey D. Camm, James J. Cochran, Michael J. Fry, Jeffrey W. Ohlmann, David R. Anderson
Publisher: Cengage Learning
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Chapter 6, Problem 14P

a.

To determine

Obtain the sampling distribution of p¯

a.

Expert Solution
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Answer to Problem 14P

The sampling distribution of p¯ for a sample of 300 is normal with mean 0.42 and standard deviation 0.0285.

Explanation of Solution

It is given that the sample proportion of doctors who think their patients receive unnecessary medical care is p=0.42 and the sample size n=300

The sampling distribution of the proportion is approximately normal if np5  and n(1p)5.

Verify the conditions:

np=300×0.42=1265

And n(1p)=300×(10.42)=1745

The condition is satisfied. Therefore, the sampling distribution of the proportion is normal.

The mean of the p¯ is E(p¯)=p  and standard deviation of p¯ is σp¯=p(1p)n

In this context, p¯ is the sample proportion of doctors who think their patients receive unnecessary medical care.

The mean of p¯ is E(p¯)=p=0.42

The standard deviation of p¯ is σp¯=p(1p)n=0.42×0.58300=0.0285

Thus, the sampling distribution of the proportion p¯ of doctors who think their patients receive unnecessary medical care is normal with mean E(p¯)=0.42 and standard deviation σp¯=0.0285

b.

To determine

Obtain the sampling distribution of p¯

b.

Expert Solution
Check Mark

Answer to Problem 14P

The sampling distribution of p¯ for a sample of 500 is normal with mean 0.42 and standard deviation 0.0221.

Explanation of Solution

It is given that the sample proportion of doctors who think their patients receive unnecessary medical care is p=0.42 and the sample size is given as n=500

The sampling distribution of the proportion is approximately normal if np5  and n(1p)5.

Verify the conditions:

np=500×0.42=2105

And n(1p)=500×(10.42)=2905

The conditions are satisfied. Therefore, the sampling distribution of the proportion is normal.

The mean of the p¯ is E(p¯)=p  and standard deviation of p¯ is σp¯=p(1p)n

Mean and standard deviation are computed as below.

E(p¯)=p=0.42

σp¯=p(1p)n=0.42×0.58500=0.0221

Thus, for a random sample of 500 doctors, the sampling distribution of the proportion p¯ of doctors who think their patients receive unnecessary medical care is normal with mean E(p¯)=0.42 and standard deviation σp¯=0.0221

c.

To determine

Obtain the sampling distribution of p¯

c.

Expert Solution
Check Mark

Answer to Problem 14P

The sampling distribution of p¯ for a sample of 1000 is normal with mean 0.42 and standard deviation 0.0156.

Explanation of Solution

It is given that the sample proportion of doctors who think their patients receive unnecessary medical care is p=0.42 and the sample size is given as n=1000

The sampling distribution of the proportion is approximately normal if np5  and n(1p)5.

Verify the conditions:

np=1000×0.42=4205

And n(1p)=1000×(10.42)=5805

The conditions are satisfied. Therefore, the sampling distribution of the proportion is normal.

The mean of the p¯ is E(p¯)=p  and standard deviation of p¯ is σp¯=p(1p)n

Mean and standard deviation of p¯ are computed as below.

E(p¯)=p=0.42

σp¯=p(1p)n=0.42×0.581000=0.0156

Thus, for a random sample of 1000 doctors, the sampling distribution of the proportion p¯ of doctors who think their patients receive unnecessary medical care is normal with mean E(p¯)=0.42 and standard deviation σp¯=0.0156

d.

To determine

Establish the smallest standard error of p¯ obtained.

d.

Expert Solution
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Explanation of Solution

The standard error of p¯ for a sample of 300 doctors, computed in part (a) is 0.0285

The standard error of p¯ for a sample of 500 doctors computed in part (b) is 0.0221

The standard error of p¯ for a sample of 1000 doctors computed in part (c) is 0.0156

It can be noted that standard error of p¯ is lowest in part (c) and it can also be observed that p¯ is same in all the parts and the sample size is largest in part (c).

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Chapter 6 Solutions

Essentials of Business Analytics (MindTap Course List)

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