Question

Suppose x has a distribution with μ = 15 and σ = 7.

(a) If a random sample of size n = 49 is drawn, find μ_x, σ_x and P(15 ≤ x ≤ 17). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(15 ≤ x ≤ 17) =

(b) If a random sample of size n = 55 is drawn, find μ_x, σ_x and P(15 ≤ x ≤ 17). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(15 ≤ x ≤ 17) =

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

Answer 1

Solution :

Given that ,

mean = = 15

standard deviation = = 7

n = 49

= 15

=  / n= 7/ 49=1

P(15≤ x ≤17 ) = P[(15 -15) / 1< ( - ) / < (17 -15) /1 )]

= P( 0< Z <2 )

= P(Z <2 ) - P(Z <0 )

Using z table

=0.9772 -0.5

=0.4772

probability=0.4772

(b)n=55

= 15

=  / n= 7/ 55=0.94

P(15≤ x ≤17 ) = P[(15 -15) / 0.94< ( - ) / < (17 -15) /0.94 )]

= P( < Z <2.13 )

= P(Z <2.13 ) - P(Z <0 )

Using z table

=0.9834 -0.5

=0.4834

probability=0.4834