Question

1) The number of times that student takes an A class, X(X has a line under) has the discrete uniform pmf: p(x) = 0.25 for x = 1,2,3,4. Recall from earlier course material that this pmf has E(X)(X has a line under)=5/2 and V(X)(X has a line under)= 15/12. A random sample of 36 students will be selected and the number of times that have taken A class will be recorded.
-Determine the probability that the mean of this sample is less that 3.

2)The lysine composition is soybean meal was measured in 9 random samples resulting in a sample mean of 22.4 g/kg and standard deviation of 1.2g/kg. Construct a 2-sided 99% confidence interval on the population standard deviation. Assume that the population is normally distributed. What is the estimated of the lower bound of this confidence interval?

Answer 1

1)

By Central limit theorem, mean of the sample = E(X) = 5/2 = 2.5

Variance of the sample = V(X) / n = (15/12)/36 = 0.03472222

Standard deviation of the sample = = 0.186339

Probability that the mean of this sample is less that 3

= P( < 3)

= P[Z < (3 - 2.5)/0.186339]

= P[Z < 2.68]

= 0.9963

2)

Degree of freedom = n-1 = 9-1 = 8

Critical value of Chi Square statistic for 99% confidence interval and df = 8 are

= 21.95 and = 1.34

Lower bound of this confidence interval =

= 0.7245 g/kg

Upper bound of this confidence interval =

= 2.9321 g/kg

2-sided 99% confidence interval on the population standard deviation is,

(0.7245 g/kg ,  2.9321 g/kg)