Question

Suppose a simple random sample of size nequals49 is obtained from a population with mu equals 71 and sigma equals 7. ​(a) Describe the sampling distribution of x overbar. ​(b) What is Upper P left parenthesis x overbar greater than 72.5 right parenthesis​? ​(c) What is Upper P left parenthesis x overbar less than or equals 68.9 right parenthesis​? ​(d) What is Upper P left parenthesis 69.65 less than x overbar less than 72.65 right parenthesis​?

Answer 1

SOLUTION:

From given data,

Suppose a simple random sample of size n equals 49 is obtained from a population with mu equals 71 and sigma equals 7.

Given that ,

mean = = 71

standard deviation = = 7

n = 49

(a)Describe the sampling distribution of x over bar.

= = 71 and

= / sqrt( n ) = 7 / sqrt (49) = 7 / 7 = 1

(b)What is Upper P left parenthesis x over bar greater than 72.5 right parenthesis​?

P( > 72.5) = 1 - P( < 72.5)

= 1 - P(( - ) / < (72.5 - 71) / 1)

= 1 - P(z < 1.5)

= 1 - 0.066807

= 0.9332

Probability = 0.9332

(c) What is Upper P left parenthesis x over bar less than or equals 68.9 right parenthesis​?

P( < 68.9) = P(( - ) / < (68.9 - 71) / 1)

P( < 68.9) = P(z < -2.1/ 1)

P( < 68.9) = P(z < -2.1)

P( < 68.9) = 0.0178

Probability = 0.0178

​(d) What is Upper P left parenthesis 69.65 less than x over bar less than 72.65 right parenthesis​?

P(69.65 < < 72.65) = P((69.65 - 71) / 1 < ( - ) / < (68.9 - 72.65) / 1 )

P(69.65 < < 72.65) = P(-1.35/1 < Z < -3.75/1)

P(69.65 < < 72.65) = P(-1.35 < Z < -3.75)

P(69.65 < < 72.65) = P(Z < -3.75) - P(Z < -1.35)

P(69.65 < < 72.65) = 0.000088 - 0.088508

P(69.65 < < 72.65) = -0.0884

Probability = -0.0884