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Chapter 7: Q. 39 (page 441)

A sample survey interviews an SRS of 267college women. Suppose (as is roughly true) that 70%of college women have been on a diet within the past12months. What is the probability that 75%or more of the women in the sample have been on a diet? Follow the four-step process.

Short Answer

Expert verified

The probability that 75%or more of the women in the sample have been on a dietP(p^鈮�75%)=0.0375.

Step by step solution

01

Given Information

Given

p=70%=0.70

p^=75%=0.75

n=267

02

Explanation

The mean of the sampling distribution of p^is equal to the population proportion p:

渭p^=p=0.70

The standard deviation of the sampling distribution of p^is:

蟽p^=p(1-p)n=0.70(1-0.70)267鈮�0.0280449

The z-score is the value decreased by the mean, divided by the standard deviation:

z=x-渭蟽=0.75-0.700.0280449鈮�1.78

Determine the corresponding probability using table A:

P(p^鈮�75%)=P(z>1.78)=P(Z<-1.78)=0.0375

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Most popular questions from this chapter

According to the U.S. census, the proportion of adults in a certain county who owned their own home was 0.71. An SRS of 100adults in a certain section of the county found that65 owned their home. Which one
of the following represents the approximate probability of obtaining a sample of 100adults in which fewer than 65own their home, assuming that this section of the county has the same overall proportion of adults who own their home as does the entire county?

(a) 10065(0.71)65(0.29)35

(b) 10065(0.29)65(0.71)35
(c) Pz<0.65-0.71(0.71)(0.29)100

(d) Pz<0.65-0.71(0.65)(0.35)100
(e) Pz<0.65-0.71(0.71)(0.29)100

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Suppose we had taken samples of size 20instead of size 10. Would the spread of the sampling distribution be larger, smaller, or about the same? Justify your answer

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(a) What is the probability that a single student randomly chosen from all those taking the test scores 23or higher? Show your work.

(b) Now take an SRS of 50students Who took the fest. What is the probability that the mean score xof these students is 23or higher? Show your work

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Do you drink the cereal milk? A USA Today Poll asked a random sample of 1012 U.S. adults what they do with the milk in the bowl after they have eaten the cereal. Of the respondents, 67% said that they drink it. Suppose that 70% of U.S. adults actually drink the cereal milk. Let p^ be the proportion of people in the sample who drink the cereal milk.

(a) What is the mean of the sampling distribution of p^ ? Why?

(b) Find the standard deviation of the sampling distribution of p^. Check to see if the 10% condition is met.

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