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City R is a large city with 4 million residents, and City S is a smaller city with 0.25 million residents. Researchers believe that the proportion of City S residents who regularly ride bicycles is between 10 percent and 25 percent and the proportion of City R residents who regularly ride bicycles is between 20 percent and 50 percent. Suppose two independent random samples of residents from each city will be taken and the proportion of residents who ride bicycles is recorded. Based on the population proportions of residents who regularly ride bicycles, which of the following sample sizes is large enough to guarantee that the sampling distribution of the difference in sample proportions will be approximately normal?
a. 30 in City R and 30 in City S.
b. 30 in City R and 60 in City S.
c. 60 in City R and 30 in City S.
d. 50 in City R and 100 in City S.
e. 100 in City R and 50 in City S.

Respuesta :

Using the Central Limit Theorem, it is found that the sample sizes that are large enough to guarantee that the sampling distribution of the difference in sample proportions will be approximately normal are given by:

e. 100 in City R and 50 in City S.

What does the Central Limit Theorem state?

It states that for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex], as long as [tex]np \geq 10[/tex] and [tex]n(1 - p) \geq 10[/tex]

In this problem, the proportions are:

[tex]p_S = 0.1, P_R = 0.2[/tex]

Since:

[tex]100(0.1) = 10[/tex]

[tex]50(0.2) = 10[/tex]

Option e is correct.

More can be learned about the Central Limit Theorem at https://brainly.com/question/24663213

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