Question

Suppose x has a distribution with μ = 27 and σ = 24. (a) If a random sample of size n = 50 is drawn, find μx, σ x and P(27 ≤ x ≤ 29). (Round σx to two decimal places and the probability to four decimal places.) μx = σ x = P(27 ≤ x ≤ 29) = (b) If a random sample of size n = 74 is drawn, find μx, σ x and P(27 ≤ x ≤ 29). (Round σ x to two decimal places and the probability to four decimal places.) μx = σ x = P(27 ≤ x ≤ 29) = (c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).) The standard deviation of part (b) is ---Select--- part (a) because of the ---Select--- sample size. Therefore, the distribution about μx is ---Select--- .

Answer 1

Solution :

Given that,

mean = = 27

standard deviation = = 24

a) n = 50

= = 27

= / n = 24 / 50 = 3.39

P(27 29)  

= P[(27 - 27) /3.39 ( - ) / (29 - 27) / 3.39 )]

= P( 0 Z 0.59)

= P(Z 0.59 ) - P(Z 0)

Using z table,  

= 0.7224 - 0.5  

= 0.2224

b) n = 74

= = 27

= / n = 24 / 74 = 2.79

P(27 29)  

= P[(27 - 27) /2.79 ( - ) / (29 - 27) / 2.79)]

= P( 0 Z 0.72)

= P(Z 0.72) - P(Z 0)

Using z table,  

= 0.7642 - 0.5  

= 0.2642

c) The standard deviation of part (b) is smaller than part (a) because of the larger sample size. Therefore, the distribution about μx is same