Question

Suppose x has a distribution with μ = 30 and σ = 29.

(a) If a random sample of size n = 36 is drawn, find μ_x, σ_x and P(30 ≤ x ≤ 32). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(30 ≤ x ≤ 32) =

(b) If a random sample of size n = 57 is drawn, find μ_x, σ_x and P(30 ≤ x ≤ 32). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(30 ≤ x ≤ 32) =

(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--- the same as larger than smaller than part (a) because of the  ---Select--- same larger smaller sample size. Therefore, the distribution about μ_x is  ---Select--- the same wider narrower .

Answer 1

Solution :

(a)

= 30

= / n = 29 / 36 = 4.83

= P[(30 - 30) / 4.83    ( - ) /    (32 - 30) / 4.83)]

= P(0    Z 0.41)

= P(Z   0.41) - P(Z 0)

= 0.1591

(b)

= 30

= / n = 29 / 57 = 3.84

= P[(30 - 30) / 3.84    ( - ) /    (32 - 30) / 3.84)]

= P(0    Z 0.52)

= P(Z   0.52) - P(Z 0)

= 0.1985

(c)

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