Answer:
Step-by-step explanation:
Given that:
the sample proportion p = 0.39
sample size = 100
Then np = 39
Using normal approximation
The sampling distribution from the sample proportion is approximately normal.
Thus, mean [tex]\mu _{\hat p} = p = 0.39[/tex]
The standard deviation;
[tex]\sigma = \sqrt{\dfrac{p(1-p)}{n} }[/tex]
[tex]\sigma = \sqrt{\dfrac{0.39(1-0.39)}{100} }[/tex]
[tex]\sigma = 0.048[/tex]
The test statistics can be computed as:
[tex]Z = \dfrac{{\hat _{p}} - \mu_{_ {\hat p}} }{\sigma_{\hat p}}[/tex]
[tex]Z = \dfrac{0.3 - 0.39 }{0.0488}[/tex]
[tex]Z = -1. 8 4[/tex]
From the z - tables;
[tex]P (\hat p \le 0.3 ) = P(z \le -1.84)[/tex]
[tex]\mathbf{P (\hat p \le 0.3 ) = 0.0329}[/tex]
(b)
Here;
the sample proportion = 0.39
the sample size n = 400
Since np = 400 * 0.39 = 156
Thus, using normal approximation.
From the sample proportion, the sampling distribution is approximate to the mean [tex]\mu_{\hat p} = p = 0.39[/tex]
the standard deviation [tex]\sigma_{\hat p} = \sqrt{\dfrac{p(1-p)}{n} }[/tex]
[tex]\sigma_{\hat p} = \sqrt{\dfrac{0.39 (1-0.39)}{400} }[/tex]
[tex]\sigma_{\hat p} =0.0244[/tex]
The test statistics can be computed as:
[tex]Z = \dfrac{{\hat _{p}} - \mu_{_ {\hat p}} }{\sigma_{\hat p}}[/tex]
[tex]Z = \dfrac{0.3 - 0.39 }{0.0244}[/tex]
[tex]Z = -3.69[/tex]
From the z - tables;
[tex]P (\hat p \le 0.3 ) = P(z \le -3.69)[/tex]
[tex]\mathbf{P (\hat p \le 0.3 ) = 0.0001}[/tex]
(c) The effect of the sample size on the sampling distribution is that:
As sample size builds up, the standard deviation of the sampling distribution decreases.
In addition to that, reduction in the standard deviation resulted in increases in the Z score, and the probability of having a sample proportion that is less than 30% also decreases.
