the proportion of twins born in a town is ( p = 0.12 ). suppose we randomly select 100 women from this town…

the proportion of twins born in a town is ( p = 0.12 ). suppose we randomly select 100 women from this town who give birth in the next year. which of the following is the mean of the sampling distribution of ( hat{p} )?\n( mu_{hat{p}}=p = 0.12 )\n( mu_{hat{p}}=np = 100(0.12)=12 )\n( mu_{hat{p}}=1 - p = 1 - 0.12 = 0.88 )\n( mu_{hat{p}}=n(1 - p)=100(1 - 0.12)=88 )

the proportion of twins born in a town is ( p = 0.12 ). suppose we randomly select 100 women from this town who give birth in the next year. which of the following is the mean of the sampling distribution of ( hat{p} )?\n( mu_{hat{p}}=p = 0.12 )\n( mu_{hat{p}}=np = 100(0.12)=12 )\n( mu_{hat{p}}=1 - p = 1 - 0.12 = 0.88 )\n( mu_{hat{p}}=n(1 - p)=100(1 - 0.12)=88 )

Answer

Explanation:

Step1: Recall the formula for the mean of the sampling distribution of (\hat{p})

The mean of the sampling distribution of the sample proportion (\hat{p}), denoted as (\mu_{\hat{p}}), is equal to the population proportion (p). Mathematically, (\mu_{\hat{p}}=p).

Step2: Substitute the given value of (p)

We are given that (p = 0.12). So, (\mu_{\hat{p}}=0.12).

Let's check the other options:

  • For the option (\mu_{\hat{p}}=np), (np) is the mean of a binomial distribution (X) (where (X) is the number of successes, not the sample proportion (\hat{p})). Here (X) would be the number of women having twins. But we are interested in the mean of (\hat{p}=\frac{X}{n}).
  • For the option (\mu_{\hat{p}}=1 - p), this formula is not related to the mean of the sampling distribution of (\hat{p}).
  • For the option (\mu_{\hat{p}}=n(1 - p)), (n(1 - p)) is the mean of the number of failures in a binomial distribution, not relevant to the mean of (\hat{p}).

Answer:

(\mu_{\hat{p}}=p = 0.12) (the first option)