calculating probabilities from the sampling distribution of $hat{p}$\naccording to a recent poll, 22% of us…

calculating probabilities from the sampling distribution of $hat{p}$\naccording to a recent poll, 22% of us adults get an average of 7 or more hours of sleep per night. assume this is the parameter value for the population. suppose you select a simple random sample of 50 us adults and find that 10 of them get an average of 7 or more hours of sleep per night. let $hat{p}$ = the proportion in the sample who get an average of 7 or more hours of sleep per night.\nassuming the polling information is true, the probability that 20% or less of us adults get an average of 7 or more hours of sleep per night is \nif needed, use the z - table to answer the question.

calculating probabilities from the sampling distribution of $hat{p}$\naccording to a recent poll, 22% of us adults get an average of 7 or more hours of sleep per night. assume this is the parameter value for the population. suppose you select a simple random sample of 50 us adults and find that 10 of them get an average of 7 or more hours of sleep per night. let $hat{p}$ = the proportion in the sample who get an average of 7 or more hours of sleep per night.\nassuming the polling information is true, the probability that 20% or less of us adults get an average of 7 or more hours of sleep per night is \nif needed, use the z - table to answer the question.

Answer

Explanation:

Step1: Identify parameters

$p = 0.22$, $n=50$, $\hat{p}=0.20$

Step2: Calculate z - score

$z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}=\frac{0.20 - 0.22}{\sqrt{\frac{0.22\times(1 - 0.22)}{50}}}\approx - 0.38$

Step3: Find probability from z - table

$P(Z\leq - 0.38)=0.3520$

Answer:

$0.3520$