Question

Suppose that the age of students at George Washington Elementary school is uniformly distributed between 5 and 11 years old. 48 randomly selected children from the school are asked their age. Round all answers to 4 decimal places where possible.

What is the distribution of X ? X ~ U( 5 Correct, 11 Correct)

Suppose that 48 children from the school are surveyed.

Then the sampling distribution is What is the distribution of ¯ x ? ¯ x ~ N( Incorrect, Incorrect)

What is the probability that the average of 48 children will be between 8 and 8.4 years old?

Answer 1

Answer:- Suppose that the age of student at George Washington Elementary school is uniformly distributed between 5 and 11 years old.

Given that:-

• n = 48

a) X ~ Uniform (a , b)

=>  Given :- (a = 5) and (b = 11)

So, X ~ Uniform(5 , 11).

b) Supposed that 48 children from the school are surveyed, then the sampling distribution is:-

=> μX̄ = (a + b)/2

= (5 + 11)/2

= 16/2

= 8

[ μX̄ = 8 ]

=> S.D (σ) = √(b - a)^2/12

= √(11 - 5)^2/12

= √(6)^2/12

= √36/12

= √3

= 1.73

[ σ = 1.73 ]

=> σX̄ = σ/√n

= 1.73/√48

= 1.73/6.928

= 0.25

[ σX̄ = 0.25 ]

So,  X̄ ~ N = [μX̄ = 8,  σX̄ = 0.25]

c) The probability that the average of 48 children will be between 8 and 8.4 years old..

=> P(8 < X̄ < 8.4):-

= P[ (8 - 8)/0.25 < Z < (8.4 - 8 )/0.25 ]

= P[ 0 < Z < (0.4)/0.25 ]

= P(0 < Z < 1.6)

= P(Z < 1.6) - P(Z < 0)

(From Z table)

= 0.9525 - 0.5

P(8 < X̄ < 8.4) = 0.4525