Question

A recent study reported that

59%

of the children in a particular community were overweight or obese. Suppose a random sample of

300

public school children is taken from this community. Assume the sample was taken in such a way that the conditions for using the Central Limit Theorem are met. We are interested in finding the probability that the proportion of​ overweight/obese children in the sample will be greater than

0.55

Complete parts​ (a) and​ (b) below.

a. Before doing any​ calculations, determine whether this probability is greater than​ 50% or less than​ 50%. Why?

A.The answer should be less than​ 50%, because 0.55 is less than the population proportion of

0.59 and because the sampling distribution is approximately Normal.

B.The answer should be greater than​ 50%, because

0.55 is less than the population proportion of

0.59 and because the sampling distribution is approximately Normal.

C.

The answer should be greater than​ 50%, because the resulting​ z-score will be positive and the sampling distribution is approximately Normal.

D.

The answer should be less than​ 50%, because the resulting​ z-score will be negative and the sampling distribution is approximately Normal.

b. Calculate the probability that

55%

or more of the sample are overweight or obese.

Upper P left parenthesis ModifyingAbove p with caret greater than or equals 0.55 right parenthesisPp≥0.55equals=nothing

​(Round to three decimal places as​ needed.)

Answer 1

The population proportion is 0.59

To find P(p > 0.55), n = 300

(a) Case 1: When sample proportion is greater than population proportion: We will have a positive z score and then the probability of getting a value lesser than the sample proportion will be more than 50% and the probability of getting a value greater than the sample proportion will be less than 50%.

Case 2: When sample proportion is lesser than population proportion: We will have a negative z score and then the probability of getting a value lesser than the sample proportion will be less than 50% and the probability of getting a value greater than the sample proportion will be more than 50%

In this case we have a sample proportion which is lesser than the population proportion and we need to find the probability that the sample proportion is > than the population proportion.

Therefore Option B: The answer should be greater than 50%because 0.55 is less than the population proportion of 0.59 and because the sampling distribution is approximately normal.

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For P(p > 0.55) = 1 - P(p < 0.55)

For P(p < 0.55)

The probability for z (-1.41) = 0.0793

Therefore the required probability P(p > 0.55) = 1 - 0.0793 = 0.9207

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