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 her students is higher than the proportion reported. She randomly surveys 90 of the 3600 students in the district and finds that 38 reported being engaged in school. Complete parts (a) through (e) below. 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. Thesuperintendent of one high school district believes the engagement of her students is higher than the proportion reported. She randomly surveys 90 of the 3600 students in the districtand finds that 38 reported being engaged in school. Complete parts (a) through (e) below.(a) What type of variable is "engaged"?The variable is(b) What is the sample proportion of students in this high school district who are engaged?p=(Round to four decimal places as needed.)(c) Verify that the distribution of the sample proportion is approximately normal.5% of the population size, or 0.05N =The sample size, n =, is(Type integers or decimals. Do not round.)10.For this sample, np (1-P) = [(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 thedistribution of the sample proportion.μ^ =...0A =(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 sampleproportion from part (d). Interpret this probability.Ppz(Round to three decimal places as needed.)Interpret this probability. Select the correct choice below and fill in the answer box to complete your hoice.(Round to the nearest whole number as needed.)OA. If 100 different random samples of 90 students were obtained, at mostwill have at least 38 students that are engaged.OB. If 100 different random samples of 90 students were obtained, at least will have at least 38 students that are engaged.OC. If 100 different random samples of 90 students were obtained, would be expected to have at least 38 students that are engaged.

Given information:Population proportion: Total population size: Sample size: The number of students engaged in the school is .a) The "engaged" variable has two possible outcomes "Yes" or "No". So, the variable has two categories. Therefore, the variable "engaged" is a categorical variable. It is also called a qualitative variable.b) The sample proportion of students who are engaged in the school is calculated as follows:Therefore, the sample proportion is .c) The sample size is good if it is less than 5% of the population size.The distribution of the sample proportion is approximately normal if .Here, the sample size, is less than the 5% of the population size. That is, For this sample, So, .Therefore, the distribution of the sample proportion is approximately normal.d) For the large sample size, the distribution of the sample proportion is approximately normal.So, the mean of the distribution of sample proportion is, The standard deviation of the sample proportion is,e) We have, .For the 38 out of 90, the proportion is 0.4222.The required probability is calculated below:The normally distributed sample proportion is converted into the standard normal using formula .The is calculated using the Excel function as,Interpretation of probability:C. If 100 different samples of 90 students were obtained, 50% would be expected to have at least 38 students that are engaged.a) The variable "engaged" is a categorical variable. It is also called a qualitative variable.b) The sample proportion is .c) The sample size, is less than 5% of the population size. For this sample, . The distribution of the sample proportion is approximately normal.d) e) The required probability is .Interpretation of probability:C. If 100 different samples of 90 students were obtained, 50% would be expected to have at least 38 students that are engaged.