Given a population mean of 27.1 with a standard deviation of 8.4 and a sample size of 10, answer these questions. part A: what is the standard deviation of the sampling distribution of x hat? show your work.
PartB: what does the sample size need to be if you want the standard deviation of the sampling distribution of x hat to be 2.8? show your work.

a) 2.66

b) 9

Step-by-step explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

part A: what is the standard deviation of the sampling distribution of x hat? show your work.

We have that . So

PartB: what does the sample size need to be if you want the standard deviation of the sampling distribution of x hat to be 2.8?

This is n when . So

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt

your answer is c. 97 (don't delete this again)

step-by-step explanation: