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 0.03 inch. A random sample of 11 tennis balls is selected.

The probability is 69% that the sample mean will be between what two values symmetrically distributed around the population​ mean? (Round to two decimal places).

The lower bound is ___ inches, the upper bound is ___ inches.

Answer 1

Solution :

Given that ,

mean = = 2.58 inches

standard deviation = = 0.03 inches

n = 11

=  2.58 inches

= / n = 0.03 / 11 = 0.01

Using standard normal table,

P( -z < Z < z) = 69%

= P(Z < z) - P(Z <-z ) = 0.69

= 2P(Z < z) - 1 = 0.69

= 2P(Z < z) = 1 + 0.69

= P(Z < z) = 1.69 / 2

= P(Z < z) = 0.0.845

= P(Z < 1.02) = 0.845

= z  ± 1.02

Using z-score formula  

= z * +

= -1.02 * 0.01 + 2.58

= 2.57 inches.

Using z-score formula  

= z * +

= 1.02 * 0.01 + 2.58

= 2.59 inches.

The lower bound is 2.57 inches, the upper bound is 2.59 inches.