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Chapter 5: Q38E (page 320)

A random sample of n = 80 measurements is drawn from a binomial population with a probability of success .3.

  1. Give the mean and standard deviation of the sampling distribution of the sample proportion, p¯
  2. Describe the shape of the sampling distribution of p¯
  3. Calculate the standard normal z-score corresponding to a value of p¯=0.35
  4. FindP(p)¯=0.35

Short Answer

Expert verified

A random sample is a subgroup of people chosen at random by investigators to represent the full population as a whole. A technique for selecting a sample of data from a community to make predictions regarding the community.

Step by step solution

01

 Step 1: (a) The data is given below

The calculation is given below:

Given,

n=80

p=0.3

Meanp= 0.3σP⏜=pqn=0.3×1-0.380=0.0512

≃0.05

02

(b) The data is given below

The calculation is given below:

np = 80×0.3 = 24⩾10nq = 80×0.7 = 56⩾10

Sincenp⩾10 and nq⩾10,the sample probability is normal distribution is normally distributed with μP⏜=0.3 andσP⏞= 0.05

03

(c) The data is given below

The calculation is given below:

For P⏞=0.35, the Z score is:

Z=P⏞-μP⏞σP⏞=0.35-0.30.05=1

04

(d) The data is given below

The calculation is given below:

PP⏜>.35=PP⏞-μP⏜σP⏞⩾0.35-μp⏜σp⏜=PZ⩾1=1-PZ<1

=1-0.8413=0.1587

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Most popular questions from this chapter

Question:Stock market participation and IQ. Refer to The Journal of Finance (December 2011) study of whether the decision to invest in the stock market is dependent on IQ, Exercise 3.46 (p. 182). The researchers found that the probability of a Finnish citizen investing in the stock market differed depending on IQ score. For those with a high IQ score, the probability is .44; for those with an average IQ score, the probability is .26; and for those with a low IQ score, the probability is .14.

a. In a random sample of 500 Finnish citizens with high IQ scores, what is the probability that more than 150 invested in the stock market?

b. In a random sample of 500 Finnish citizens with average IQ scores, what is the probability that more than 150 invest in the stock market?

c. In a random sample of 500 Finnish citizens with low IQ scores, what is the probability that more than 150 invest in the stock market?

Dentists’ use of laughing gas. According to the American Dental Association, 60% of all dentists use nitrous oxide (laughing gas) in their practice. In a random sample of 75 dentists, let p^represent the proportion who use laughing gas in practice.

a. Find Ep^.

b. Find σp^.

c. Describe the shape of the sampling distribution of p^.

d. Find Pp^>0.70.

Exposure to a chemical in Teflon-coated cookware. Perfluorooctanoic acid (PFOA) is a chemical used in Teflon-coated cookware to prevent food from sticking. The EPA is investigating the potential risk of PFOA as a cancer-causing agent (Science News Online, August 27, 2005). It is known that the blood concentration of PFOA in people in the general population has a mean of parts per billion (ppb) and a standard deviation of ppb. Science News Online reported on tests for PFOA exposure conducted on a sample of 326 people who live near DuPont’s Teflon-making Washington (West Virginia) Works facility.

a. What is the probability that the average blood concentration of PFOA in the sample is greater than 7.5 ppb?

b. The actual study resulted in x¯=300ppb. Use this information to make an inference about the true meanμPFOA concentration for the population of people who live near DuPont’s Teflon facility.

Suppose a random sample of n measurements is selected from a population with u=100mean and variance role="math" localid="1657967387987" σ2=100. For each of the following values of n, give the mean and standard deviation of the sampling distribution of the sample mean.

  1. role="math" localid="1657967260825" n=4
  2. n=25
  3. n=100
  4. n=50
  5. n=500
  6. n=1000

Refer to Exercise 5.18. Find the probability that

  1. x¯is less than 16.
  2. x¯is greater than 23.
  3. x¯is greater than 25.
  4. x¯falls between 16 and 22.
  5. x¯is less than 14.
See all solutions

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