Question

Suppose x has a distribution with μ = 10 and σ = 3.

(a) If a random sample of size n = 47 is drawn, find μx, σ x and P(10 ≤ x ≤ 12). (Round σx to two decimal places and the probability to four decimal places.)

μx =

σ x =

P(10 ≤ x ≤ 12) =

(b) If a random sample of size n = 58 is drawn, find μx, σ x and P(10 ≤ x ≤ 12). (Round σ x to two decimal places and the probability to four decimal places.)

μx =

σ x =

P(10 ≤ x ≤ 12) =

(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 part (a) because of the sample size. Therefore, the distribution about μx is

Answer 1

Solution :

Given that ,

a) mean = = 10

standard deviation = = 3

n = 47

= = 10

= / n = 3 / 47 = 0.44

P( 10 x 12)

= P[(10 -10 /0.44 ) (x - ) / (12 -10 /0.44 ) ]

= P(0 z 4.55 )

= P(z 4.55) - P(z 0)

= 1 - 0.5 = 0.5

probability = 0.5000

b)

n = 58

= = 10

= / n = 3 / 58 = 0.40

P( 10 x 12)

= P[(10 -10 /0.40 ) (x - ) / (12 -10 /0.40) ]

= P(0 z 5 )

= P(z 5) - P(z 0)

= 1 - 0.5 = 0.5

probability = 0.5000

c) because the sample size increased standard deviation decreased.