Respuesta :
Answer:
The probability that the value of P will be less than 0.81 is 0.7642.
Step-by-step explanation:
According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.
The mean of this sampling distribution of sample proportion is:
[tex]\mu_{\hat p}=p[/tex]
The standard deviation of this sampling distribution of sample proportion is:
[tex]\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}[/tex]
Given:
n = 100
p = 0.78
Since n = 100 > 30, according to the central limit theorem the sampling distribution of sample proportion follows a Normal distribution.
[tex]\hat p\sim N(\mu_{\hat p}=0.78, \sigma_{\hat p}=0.0414)[/tex]
Compute the value of [tex]P(\hat p<0.81)[/tex] as follows:
[tex]P(\hat p<0.81)=P(\frac{\hat p-\mu_{\hat p}}{\sigma_{\hat p}}<\frac{0.81-0.78}{0.0414})[/tex]
[tex]=P(Z<0.72)\\=0.76424\\\approx0.7642[/tex]
*Use a z-table for the probability.
Thus, the probability that the value of P will be less than 0.81 is 0.7642.
