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)
(a) what type of variable is \engaged\?
the variable is qualitative
(b) what is the sample proportion of students in this high school district who are engaged?
\\( \hat { p } = 0.4222 \\)
(round to four decimal places as needed.)
(c) verify that the distribution of the sample proportion is approximately normal
the sample size, \\( n = \square \\), is less than 5% of the population size, or \\( 0.05 n = \square \\)
(type integers or decimals do not round)

Explanation:

Step1: Identify sample size and population size

The sample size \(n = 90\) (given as the number of students surveyed). The population size \(N=3600\) (total number of students in the district).

Step2: Calculate \(0.05N\)

$$0.05N=0.05\times3600 = 180$$

Answer:

The sample size, \(n = 90\), is less than \(5\%\) of the population size, or \(0.05N = 180\)