Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
3rd Edition
ISBN: 9781259969454
Author: William Navidi Prof.; Barry Monk Professor
Publisher: McGraw-Hill Education
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Chapter 7.4, Problem 22E

Blood pressure: High blood pressure has been identified as a risk factor for heart attacks and strokes. The National Health and Nutrition Examination Survey reported that the proportion of U.S. adults with high blood pressure is 0.3. A sample of 38 U.S. adults is chosen.

  1. Is it appropriate to use the normal approximation to find the probability that more than 40% of the people in the sample have high blood pressure? If so, find the probability. If not; explain why not.
  2. A new sample of 80 adults is drawn. Find the probability that more than 40% of the people in this sample have high blood pressure.
  3. Find the probability that the proportion of individuals in the sample of 80 who have high blood pressure is between 0.20 and 0.35.
  4. Find the probability that less than 25% of the people in the sample of 80 have high blood.
  5. Would it be unusual if more than 45% of the individuals in the sample of 80 had high blood pressure?

a.

Expert Solution
Check Mark
To determine

Whether it is appropriate to use the normal approximation to find the probability that more than 40% of the people in the sample have high blood pressure under given scenario.

Answer to Problem 22E

The probability that more than 40% of the people in the sample have high blood pressure.

is 0.0764 .

Explanation of Solution

Given Information:

The National Health and Nutrition Examination Survey reported that the proportion of U.S. adults with high blood pressure is 0.3. A sample of 38 U.S. adults is chosen.

Formula used:

The standard deviation is

  σP=p( 1p)n

where n is sample size and p is population proportion.

Calculation:

Given that the sample size is n=38 and population proportion is p=0.3 .

Since, np=(38)(0.3)=11.410

And n(1p)=(38)(10.3)=26.610 .

This means normal approximation can be used to find the probability that more than 40% of the people in the sample have high blood pressure.

The mean is μp=p=0.3

The standard deviation is,

  σp= p( 1p )n= 0.3( 10.3 ) 38=0.07

Calculating the z score for 0.40 as below,

  z=p^μpσp=0.400.30.07=1.43

The required probability is

  P(p^>0.40)=1P(p^<0.40)=1P(z<1.43)=10.9236=0.0764

Hence, the probability that more than 40% of the people in the sample have high blood pressure.

is 0.0764 .

b.

Expert Solution
Check Mark
To determine

The probability of more than 40% of the people in the sample of 80 adults having high blood pressure.

Answer to Problem 22E

The probability that more than 40% of the people in the sample of 80 adults have high blood pressure is 0.0228 .

Explanation of Solution

Given Information:

  p=0.4n=80p=0.3

Formula used:

  σp^=p( 1p)n

Calculation:

Given that the sample size is n=80 and population proportion is p=0.4 .

Since, np=(80)(0.4)=3210

And n(1p)=(80)(10.4)=4810 .

This means normal approximation can be used to find the probability that more than 40% of the people in the sample have high blood pressure.

Now mean is μp=p=0.3

The standard deviation is,

  σp= p( 1p )n= 0.3( 10.3 ) 80=0.05

Calculating the z score for 0.40 as below,

  z=p^μpσp=0.400.30.05=2

The required probability will be given as,

  P(p^>0.40)=1P(p^<0.40)=1P(z<2)=10.9772=0.0228

Hence, the probability that more than 40% of the people in the sample of 80 adults have high blood pressure is 0.0228 .

c

Expert Solution
Check Mark
To determine

The probability that the proportion of individuals in the sample of 80 who have high blood pressure is between 0.20 and 0.35 .

Answer to Problem 22E

The probability that the proportion of individuals in the sample of 80 who have high blood pressure is between 0.20 and 0.35 would be 0.8185 .

Explanation of Solution

Given Information:

  p1=0.4,p2=0.4,n=80,μp=p=0.3,σp=0.05

Calculation:

Calculating the z score for 0.20 as below,

  z=p^μpσp=0.200.30.05=2

Calculating the z score for 0.35 as below,

  z=p^μpσp=0.350.30.05=1

The required probability is

  P(0.20<p^<0.35)=P(p^<0.35)P(p^<0.20)=P(z<1)P(z<2)=0.84130.0228=0.8185

Hence, the probability that the proportion of individuals in the sample of 80 who have high blood pressure is between 0.20 and 0.35 would be 0.8185 .

d

Expert Solution
Check Mark
To determine

The probability that less than 25% of the people in the sample of 80 have high blood pressure.

Answer to Problem 22E

The required answer is 0.1587

Explanation of Solution

Given Information:

  p1=0.25,n=80,μp=p=0.3,σp=0.05

Calculation:

Calculating the z score for 0.25 as below,

  z=p^μpσP=0.250.30.05=1

The required probability is

  P(p^<0.25)=P(z<1)=0.1587

Hence, the probability that less than 25% of the people in the sample of 80 have high blood pressure would be 0.1587 .

e.

Expert Solution
Check Mark
To determine

If it is unusual for more than 45% of the individuals in the sample of 80 to have high blood pressure.

Answer to Problem 22E

Yes, it would be unusual if more than 45% of the individuals in the sample of 80 had a high blood pressure.

Explanation of Solution

Given Information:

  p=0.45,μp=p=0.3,σp=0.05

Calculation:

Calculating the z score for 0.25 as below,

  z=p^μpσP=0.450.30.05=3

The required probability is.

  P(p^<0.45)=1P(z<3)=10.9987=0.0013

The above probability is very low. Hence, it would be unusual if more than 45% of the individuals in the sample of 80 had a high blood pressure.

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Chapter 7 Solutions

Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561

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