Suppose x has a distribution with a mean of 80 and a standard deviation of 3. random samples of size n = 36 are drawn. (a) describe the x distribution and compute the mean and standard deviation of the distribution. x has distribution with mean μx = and standard deviation σx = . (b) find the z value corresponding to x = 81. z = (c) find p(x < 81). (round your answer to four decimal places.) p(x < 81) = (d) would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 81? explain. yes, it would be unusual because less than 5% of all such samples have means less than 81. no, it would not be unusual because more than 5% of all such samples have means less than 81. no, it would not be unusual because less than 5% of all such samples have means less than 81. yes, it would be unusual because more than 5% of all such samples have means less than 81.

a)  

b)

c)

d) No, it would not be unusual because more than 5% of all such samples have means less than 81

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".  

Part a

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

Where and

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

On this case  

Part b

In order to find the z score we can use this formula:

If we replace on the before formula we got:

Part c

We want this probability:

Part d

If we see the probability obtained on part c we have that more than 90% of the samples have samples means less than 81. So the best answer is:

No, it would not be unusual because more than 5% of all such samples have means less than 81

answer: the first graph

step-by-step explanation:

answer:

your answer is: c. 196 7/8 in^3.