Question

The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean  and standard deviation .

(a) What is the probability that a single student randomly chosen from all those taking the test scores 545 or higher?
ANSWER:  

For parts (b) through (d), consider a simple random sample (SRS) of 30 students who took the test.

(b) What are the mean and standard deviation of the sample mean score , of 30 students?
The mean of the sampling distribution for  is:  
The standard deviation of the sampling distribution for  is:

(c) What z-score corresponds to the mean score  of 545?
ANSWER:

(d) What is the probability that the mean score  of these students is 545 or higher?
ANSWER:  

Answer 1

Dear student, values of mean and standard deviation is not given here. Let me go through the process of doing every part of this question with an assumption that the mean is and the standard deviation is .

Provide us the value, and we will solve it with the values in it.

= standard score

= observed value

= mean of the sample

= standard deviation of the sample

a)

Here, first, calculate the z-score for the value 545 by putting the value 545 in the place of x.

And then, Go to z-table and check the probability there.

Suppose if z-score comes out to be 1.66, then check 1.6 in the columns and 0.06 in the rows. The value you get will be the value to the left of the score 545(means equal and less than 545). For more than 545, subtract it with 1.

b)

The mean of the sampling distribution for is: It can be calculated by just taking the sum of all the values and dividing it by 30.
The standard deviation of the sampling distribution is: It can be calculated by using the below formula.

c)

Use the z-score formula. This time, mean and standard deviation will change, take mean, and standard deviations value which you have got in part b.

d)

Repeat the steps in part with the different mean and standard deviations value.