Question

The diameter of a brand of tennis balls is approximately normally​ distributed, with a mean of 2.58 inches and a standard deviation of .04 inch. A random sample of 11 tennis balls is selected. Complete parts​ (a) through​ (d) below.

a. What is the sampling distribution of the​ mean?

A.Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 11 will be the uniform distribution.

B.Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 11 will not be approximately normal.

C.Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 11 cannot be found.

D.Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 11 will also be approximately normal.

b. What is the probability that the sample mean is less than 2.55 ​inches? P(X<2.55​)= ​(Round to four decimal places as​ needed.)

c. What is the probability that the sample mean is between 2.57 and 2.60 ​inches? ​

P(2.57< X< 2.60)=

​(Round to four decimal places as​ needed.)

Answer 1

a)

Because the population diameter of tennis balls is approximately normally​ distributed, the sampling

distribution of samples of size 11 will also be approximately normal.

b)

Given,

= 2.58, = 0.04

Using central limit theorem,

P( < x) = P( Z < x - / ( / sqrt(n) ) )

So,

P( < 2.55) = P( Z < 2.55 - 2.58 / 0.04 / sqrt(11) )  

= P( Z < -2.4875)

= 0.0064

c)

P( 2.57 < < 2.60) = P( < 2.60) - P( < 2.57)

= P( Z < 2.60 - 2.58 / 0.04 / sqrt(11) ) - P( Z < 2.57 - 2.58 / 0.04 / sqrt(11) )

= P( Z < 1.6583) - P( Z < -0.8292)

= 0.9514 - 0.2035

= 0.7475