Over the entire six years that students attend an Ohio elementary school,they are absent,on average,28 days due to influenza.Assume that the standard deviation over this time period is σ = 9 days.Upon graduation from elementary school,a random sample of 36 students is taken and asked how many days of school they missed due to influenza.The probability that the sample mean is between 25 and 30 school days is _______.
A) 0.0228
B) 0.0918
C) 0.8854
D) 0.9082

Question

Quizplus · Accepted Answer

We are given the following information:
- Population mean (μ) = 28 days
- Standard deviation (σ) = 9 days
- Sample size (n) = 36
We need to find the probability that the sample mean (x̄) is between 25 and 30 school days.
We can use the central limit theorem to approximate the distribution of sample means as a normal distribution with mean (μx̄) = μ = 28 days and standard deviation (σx̄) = σ/√n = 9/√36 = 1.5 days.

P(25 ≤ x̄ ≤ 30) can be standardized to Z-score

z1= (25-28)/1.5 = -2

z2= (30-28)/1.5 = 1.33

The probability of z= -2 upto z = 1.33 will give the probability that the sample mean is between 25 and 30 days

Using standard normal table, we can find:

P(-2 ≤ Z ≤ 1.33) = P(Z ≤ 1.33) - P(Z ≤ -2)
= 0.9082 - 0.0228
= 0.8854

Therefore, the probability that the sample mean is between 25 and 30 school days is 0.8854. The best choice is (C).

Over the Entire Six Years That Students Attend an Ohio