In a data set with a normal distribution, the mean is 59 and the standard deviation is 3.

Part A

About what percent of the data lie between 53 and 62?

A. 68%

B. 83.85%

C. 81.5%

D. 50%

Part B

Complete: About 95% of the data lie between

a)

b)

And for the other limit we have:

And we will have 95% of the values between (53.12, 64.88)

Step-by-step explanation:

Part a

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

Where and

We are interested on this probability :

And we can solve this using the z score:

Using the last formula we got:

And we can find this probability with this difference:

And uing the normal standard distribution or excel we got:

Part b

For this part we want to find a value a, such that we satisfy this condition:

  (a)

  (b)

As we can see on the figure attached the z value that satisfy the condition with 0.95 of the area on the left and 0.05 of the area on the right it's z=1.96.

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

And for the other limit we have:

And we will have 95% of the values between (53.12, 64.88)

y= -