Question

The diameter of a brand of tennis balls is approximately normally​ distributed, with a mean of

2.69

inches and a standard deviation of

0.05

inch. A random sample of

10

tennis balls is selected. Complete parts​ (a) through​ (d) below.

b.

What is the probability that the sample mean is less than

2.68 ​inches?

c.What is the probability that the sample mean is between

2.67 and 2.70 ​inches?

d.  The probability is 71% that the sample mean will be between what two values symmetrically distributed around the population​ mean?

e. The probability is 51% that the sample mean will be between what two values symmetrically distributed around the population​ mean?

Answer 1

Solution :

Given that ,

mean = = 2.69

standard deviation = = 0.05

n = 10

= 2.69

= / n = 0.05 / 10 = 0.0158

b.

P( < 2.68) = P(( - ) / < (2.68 - 2.69) /0.0158 )

= P(z < -0.63)

= 0.2643

probability = 0.2643

c.

= P[(2.67 - 2.69) /0.0158 < ( - ) / < (2.70 - 2.69) / 0.0158)]

= P(-1.27 < Z < 0.63)

= P(Z < 0.63) - P(Z < -1.27)

= 0.7357 - 0.1020

= 0.6337

probability = 0.6337

d.

P(-z Z z) = 71%

P(Z z) - P(Z -z) = 0.71

2P(Z z) - 1 = 0.71

2P(Z z) = 1 + 0.71= 1.71

P(Z z) = 1.71 / 2 = 0.855

P(Z 1.06) = 0.855

Z = -1.06

Using z-score formula,

= z * + = -1.06 * 0.0158 + 2.69 = 2.67

and

= 1.06 * 0.0158 + 2.69 = 2.71

Two values symmetrically distributed around the population​ mean is 2.67 and 2.71

e.

P(-z Z z) = 51%

P(Z z) - P(Z -z) = 0.51

2P(Z z) - 1 = 0.51

2P(Z z) = 1 + 0.51 = 1.51

P(Z z) = 1.51 / 2 = 0.755

P(Z 0.69) = 0.755

Z = 0.69

Using z-score formula,

= z * + = -0.69 * 0.0158 + 2.69 = 2.68

and

= 0.69 * 0.0158 + 2.69 = 2.70

Two values symmetrically distributed around the population​ mean is 2.68 and 2.70