Answer:
a) The mean of a sampling distribution of  is
 is  . The standard deviation is
. The standard deviation is  .
. 
b) The standard normal z-score corresponding to a value of  is
 is  .
. 
c) The standard normal z-score corresponding to a value of  is
 is  .
.
d) The probability  .
. 
e) The probability  .
.
f)   .
. 
Step-by-step explanation:
We are dealing here with the concept of <em>a sampling distribution</em>, that is, the distribution of the sample means  .
.
We know that for this kind of distribution we need, at least, that the sample size must be  observations, to establish that:
 observations, to establish that:
 
 
In words, the distribution of the sample means follows, approximately, a <em>normal distribution</em> with mean,  , and standard deviation (called <em>standard error</em>),
, and standard deviation (called <em>standard error</em>),  .
.
The number of observations is n = 64.
We need also to remember that the random variable Z follows a <em>standard normal distribution</em> with  and
 and  .
.

The variable Z is
 [1]
 [1]
With all this information, we can solve the questions.
Part a
The mean of a sampling distribution of  is the population mean
 is the population mean  or
 or  .
.
The standard deviation is the population standard deviation  divided by the root square of n, that is, the number of observations of the sample. Thus,
 divided by the root square of n, that is, the number of observations of the sample. Thus,  .
.
Part b
We are dealing here with a <em>random sample</em>. The z-score for the sampling distribution of  is given by [1]. Then
 is given by [1]. Then





Then, the <em>standard normal z-score</em> corresponding to a value of  is
 is  .
.
Part c
We can follow the same procedure as before. Then


 
 
 
 
 
 
As a result, the <em>standard normal z-score</em> corresponding to a value of  is
 is  .
.
Part d
Since we know from [1] that the random variable follows a <em>standard normal distribution</em>, we can consult the <em>cumulative standard normal table</em> for the corresponding  already calculated. This table is available in Statistics textbooks and on the Internet. We can also use statistical packages and even spreadsheets or calculators to find this probability.
 already calculated. This table is available in Statistics textbooks and on the Internet. We can also use statistical packages and even spreadsheets or calculators to find this probability.
The corresponding value is Z = -2, that is, it is <em>two standard units</em> <em>below</em> the mean (because of the <em>negative</em> value). Then, consulting the mentioned table, the corresponding cumulative probability for Z = -2 is  .
.
Therefore, the probability  .
.
Part e
We can follow a similar way than the previous step.

For  using the <em>cumulative standard normal table</em>, we can find this probability knowing that
 using the <em>cumulative standard normal table</em>, we can find this probability knowing that


Thus


Therefore, the probability  .
.
Part f
This probability is  and
 and  .
. 
For finding this, we need to subtract the cumulative probabilities for  and
 and 
Using the previous <em>standardized values</em> for them, we have from <em>Part d</em>:
 
 
We know from <em>Part e</em> that




Therefore,  .
.