A simple random sample of size n=1320 is obtained from a population whose size is N=1,400,000 and whose population proportion with a specified characteristic is p=0.46 . Describe the sampling distribution of $\hat { p }$.
A) Approximately normal; $\mu _ { \mathrm { p } } = 0.46 , \sigma _ { \mathrm { p } } = 0.014$.
B) Exactly normal; $\mu _ { \mathrm { p } } = 0.46 , { } _ { \mathrm { p } } \mathrm { p } = 0.1332$
C) Approximately normal; $\mu _ { \mathrm { p } } = 0.46 , \sigma _ { \mathrm { p } } = 0.014$
D) Exactly normal; $\mu _ { p } = 0.46 , \sigma p = 0.1332$

Question

A simple random sample of size n=1320  is obtained from a population whose size is  N=1,400,000  and whose population proportion with a specified characteristic is p=0.46 . Describe the sampling distribution of $\hat { p }$.
A) Approximately normal; $\mu _ { \mathrm { p } } = 0.46 , \sigma _ { \mathrm { p } } = 0.014$.
B) Exactly normal; $\mu _ { \mathrm { p } } = 0.46 , { } _ { \mathrm { p } } \mathrm { p } = 0.1332$
C) Approximately normal; $\mu _ { \mathrm { p } } = 0.46 , \sigma _ { \mathrm { p } } = 0.014$
D) Exactly normal; $\mu _ { p } = 0.46 , \sigma p = 0.1332$

Quizplus · Accepted Answer

The sampling distribution of $\hat { p }$ is approximately normal due to the large sample size (n=1320), with a mean ($\mu _ { \mathrm { p } }$) of 0.46 (the population proportion) and a standard deviation ($\sigma _ { \mathrm { p } }$) calculated using the formula for the standard deviation of a sample proportion, which results in 0.014. The conditions for normality are met (np and n(1-p) are both greater than 10), but the distribution is not exactly normal due to the sample being drawn from a population without replacement, though it is a good approximation given the large population size.

A Simple Random Sample of Size N=1320 Is Obtained p^\hat { p }p^​

A Simple Random Sample of Size N=1320 Is Obtained $\hat { p }$