Question

Suppose x has a distribution with μ = 17 and σ = 15.

(a) If a random sample of size n = 42 is drawn, find μ_x, σ_x and P(17 ≤ x ≤ 19). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(17 ≤ x ≤ 19) =

(b) If a random sample of size n = 74 is drawn, find μ_x, σ_x and P(17 ≤ x ≤ 19). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(17 ≤ x ≤ 19) =

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

Answer 1

Solution :

Given that,

mean = = 17

standard deviation = = 15

a) n = 42

= = 17

= / n = 15 / 42 = 2.31

P(17 19)  

= P[(17 - 17) / 2.31 ( - ) / (19 - 17) / 2.31)]

= P( 0 Z 0.87)

= P(Z 0.87 ) - P(Z 0)

Using z table,  

= 0.8078 - 0.5

= 0.3078

b) n = 74

= = 17

= / n = 15 / 74 = 1.74

P(17 19)  

= P[(17 - 17) / 1.74 ( - ) / (19 - 17) / 1.74)]

= P( 0 Z 1.15)

= P(Z 1.15) - P(Z 0)

Using z table,  

= 0.8749 - 0.5

= 0.3749

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