Can someone please help me out with this statistics practice problem, thank you?

Question

A manager believes that the average sales volume per customer is distributed with a population mean of $100 and a standard deviation of $35. A sample of 50 customers resulted in a sample mean of $90. What is the probability of obtaining a sample with a sample mean smaller than $90?

Answer 1

You may assume that what the question is asking is given the hypothesis that the manager's assumption is true, what is the probability of obtaining a sample mean of 90 or less?

If the probability is low then you would consider rejecting the null in favor of a mean lower than 100.

the sample dev of the sample mean is 35/sqrt(50)
we assume the mean is normally distributed, and so we standardize the mean to get a z value.
z = (90-100)/(35/sqrt(50)) = -2.02

P(Z<= -2.02) = 0.0217

Answer 2

You are given the population parameters, M = $100 and SD = $35; so you can find the area under the standard normal curve from z < -3 up to, but not including, z = (s - M)/SD. I've set z = 0 as the average for the standard normal curve; so that z < -3 is the extreme left hand tail.

You may need to convert these z values to whatever table you'll be using to pick out the areas. For example, because the curve is Normal, it is symmetric around the average z. So many tables start with P(z >= 0) = .5 just to avoid working with minus signs.

By going into the standard Normal curve tables you can look up the area between the extreme left hand tail (less than minus three standard deviations from the z = 0 point) and the point on the SNC that is equivalent to s < $90. That point is z(s < 90) < (s - M)/SD < (90 - 100)/35 < -10/35 < -.2857

So look up the area for z < -3 and then for z < -.2857; the area difference between them is the probability P(-3 < z < -.2857) = P(z < -.2857) = P(s < $90)

Answer 3

This distribution has a mean of 100 and sd =35

SE of the mean = 35/sq.rt 100 = 3,5
The sample mean was 90, ie 10/ 3.5 = 2.857 sigma from the population mean.
Referring to tables, the probability of a sample mean smaller than this is 0.001, or 0.1%

Answer 4

assume this is normal distribution, the answer is = normdist(90, 100, 35, 0) = 1.09%

Answer 5

Start by finding the z score for 90
After you do that, use your z score table to find the area to the left of your answer. That decimal will be your probability. I would help more but my calculator just died.