Question

a polling agency reported that the proportion of 12th grade students who are engaged is 0.34. engagement is defined as students involvement and enthusiasm with school. the superintendent of one high school district believes the engagement of the districts students is higher than the proportion reported. the district randomly surveys 90 of the 3600 students in the district and finds that 38 reported being engaged in school. complete parts (a) through (e)
(c) verify that the distribution of the sample proportion is approximately normal.
the sample size, ( n = 90 ), is less than 5% of the population size, or ( 0.05n = 180 )
(type integers or decimals. do not round.)
for this sample, ( nhat{p}(1 - hat{p}) = 22.0 geq 10 ).
(round to one decimal place as needed.)
(d) assuming that the proportion of students in this high school district is the same as what was reported by the polling agency, determine the mean and standard deviation of the distribution of the sample proportion.
( mu_{hat{p}} = 0.34 )
( sigma_{hat{p}} = 0.0499 )
(round to four decimal places as needed.)
(e) find the probability of obtaining a random sample of 38 or more out of 90 students who are engaged using the mean and standard deviation of the distribution of the sample proportion from part (d). interpret this probability.
( p(hat{p} geq square) = square )
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion

The sample proportion \(\hat{p}=\frac{38}{90}\approx0.422\)

Step2: Calculate the z - score

The formula for the z - score is \(z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}\)
We know that \(\mu_{\hat{p}} = 0.34\) and \(\sigma_{\hat{p}}=0.0499\)
Substitute the values: \(z=\frac{0.422 - 0.34}{0.0499}=\frac{0.082}{0.0499}\approx1.64\)

Step3: Find the probability

We want to find \(P(\hat{p}\geq0.422)\), which is equivalent to \(P(Z\geq1.64)\)
Since \(P(Z\geq z)=1 - P(Z < z)\)
From the standard normal table, \(P(Z < 1.64)=0.9495\)
So \(P(Z\geq1.64)=1 - 0.9495 = 0.0505\approx0.051\)

Answer:

\(P(\hat{p}\geq0.422)=0.051\)