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)
(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 0.422 ) = 0.051 )
(round to three decimal places as needed.)
interpret this probability. select the correct choice and fill in the answer box to complete your choice
(round to the nearest whole number as needed.)
a. for every 100 different random samples of 90 students obtained, exactly will have at least 38 students that are engaged.
b. if 100 different random samples of 90 students were obtained, would be expected to have at least 38 students that are engaged.
c. if 100 different random samples of 90 students were obtained, at least will have at least 38 students that are engaged
d. if 100 different random samples of 90 students were obtained, at most will have at least 38 students that are engaged.

Explanation:

Probability \( P(\hat{p}\geq0.422) = 0.051\) means that out of 100 samples, we expect \(100\times0.051 = 5.1\approx5\) samples to have the proportion of engaged students as \(0.422\) or higher (i.e., at least 38 out of 90 students engaged, since \(\hat{p}=\frac{38}{90}\approx0.422\)). Option B correctly interprets this as an expected - value interpretation of probability.

Answer:

B. If 100 different random samples of 90 students were obtained, 5 would be expected to have at least 38 students that are engaged.