Question

Suppose x has a distribution with μ = 12 and σ = 5.

(a) If a random sample of size n = 31 is drawn, find μ_x, σ_x and P(12 ≤ x ≤ 14). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(12 ≤ x ≤ 14) =

(b) If a random sample of size n = 67 is drawn, find μ_x, σ_x and P(12 ≤ x ≤ 14). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(12 ≤ x ≤ 14) =

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

Answer 1

Solution :

Given that,

mean = = 12

standard deviation = = 5

a) n = 31

= = 12

= / n = 5 / 31 = 0.90

P( 12 14)  

= P[(12 - 12) /0.90 ( - ) / (14 - 12) / 0.90 )]

= P( 0 Z 2.22)

= P(Z 2.22) - P(Z 0)

Using z table,  

= 0.9868 - 0.5  

= 0.4868

b) n = 67

= = 12

= / n = 5 / 67 = 0.61

P( 12 14)  

= P[(12 - 12) /0.61 ( - ) / (14 - 12) / 0.61 )]

= P( 0 Z 3.28)

= P(Z 3.28) - P(Z 0)

Using z table,  

= 0.9995 - 0.5  

= 0.4995

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