Suppose x has a distribution with a mean of 75 and a standard deviation of 45. random samples of size n = 81 are drawn.

(a) describe the x distribution and compute the mean and standard deviation of the distribution.

(b) find the z value corresponding to x = 90.

(c) find p(x < 90).

(d) would it be unusual for a random sample of size 81 from the x distribution to have a sample mean less than 90? explain.

a)    

b)

c)

d) No. it would not be unusual because more than 5% of all such samples hav means less than 90.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

(a) Describe the x distribution and compute the mean and standard deviation of the distribution.

Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:

And let represent the sample mean, the distribution for the sample mean is given by:

On this case  

(b) Find the z value corresponding to .

The z score on this case is given by this formula:

And if we replace we got:

(c) Find .

For this case we can use a table or excel to find the probability required:

(d) Would it be unusual for a random sample of size 81 from the x distribution to have a sample mean less than 90? Explain.

For this case the best conclusion is:

No. it would not be unusual because more than 5% of all such samples hav means less than 90.

c

step-by-step explanation: