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

$30.32

step-by-step explanation:

65.00*.12=2.80

65.00-2.80=57.20

57.20*1.06=60.632

60.632/2=30.316

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1