Question

Suppose x has a distribution with μ = 12 and σ = 8.

(a) If a random sample of size n = 33 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 = 61 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--- 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

X has a distribution with  μ = 12 and σ = 8.

a)

Sample size, n= 33

= 12

= = =2.3094

Let ~ Normal ( 12, 2.3094)

P( 12 <= <= 14) = P( <   < )

= P( 0 < z < 0.87)

= P( z < 0.87) - P( z< 0)

=0.80785- 0.5

= 0.30785

b)

Sample size, n= 61

= 12

= = =1.0243

Let ~ Normal ( 12, 1.0243)

P( 12 <= <= 14) = P( <   < )

= P( 0 < z < 1.95)

= P( z < 1.95) - P( z< 0)

=0.97441- 0.5

= 0.47441

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