Question

Suppose x has a distribution with μ = 12 and σ = 7.

(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 = 75 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 ______(the same as, larger than smaller than, or smaller than) part (a) because of the ______ (smaller larger, or same) sample size. Therefore, the distribution about μ_x is  _____ (the same, narrower, or wider) .

Answer 1

Solution :

Given that,

mean = = 12

standard deviation = = 7

n = 31

= 12

= / n = 7 / 12 = 1.2572

=12

= 1.2572

P( 12 14 )  

= P[( 12 - 12 ) / 1.2572 ( - ) / ( 14 - 12 ) / 1.2572 )]

= P( 0 Z 1.59 )

= P( Z 1.59 ) - P( Z 0 )

Using z table,  

= 0.9441 - 0.5

= 0.4441   

probability = 0.4441

n = 75

= 12

= / n = 7 / 75 = 0.8083

= 12

= 0.8083

P( 12 14 )  

= P[( 12 - 12 ) / 0.8083 ) ( - ) / ( 14 - 12 ) / 0.8083 )]

= P( 0 Z 2.47 )

= P( Z 2.47 ) - P( Z 0 )

Using z table,  

= 0.9932 - 0.5

= 0.4932

probability = 0.4932

The standard deviation of part (b) is _larger than_part (a) because of the _smaller_sample size. Therefore, the distribution about μx is  _the same.