Question

Noise levels at 3 airports were measured in decibels yielding the following data:

108,146,160

Required:
a. Construct the 90% confidence interval for the mean noise level at such locations. Assume the population is approximately normal.
b. Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Answer 1

a) The 90% confidence interval for the mean noise level at such locations

(112.46 , 163.54)

b) The critical value that should be used in constructing the confidence interval

Z₀.₁₀ = 1.645

Explanation:

Step(i) :-

Noise levels at 3 airports were measured in decibels yielding the following data

108 146 160

Mean of given data

`x^(-) = (108+146+160)/(3) = 138`

x : 108 146 160

x-x⁻ : -30 8 22

(x-x⁻ )² : 900 64 484

Variance σ ² = ∑(x-x⁻ )²/ n-1

=
`(900+64+484)/(3-1)= 724`

Standard deviation

σ = √724 = 26.90

Step(ii):-

The 90% confidence interval for the mean noise level at such locations

`(x^(-) - Z_(0.90) (S.D)/(√(n) ) , (x^(-) + Z_(0.90) (S.D)/(√(n) ) )`

The critical value that should be used in constructing the confidence interval

Z₀.₁₀ = 1.645

`(138 - 1.645 (26.90)/(√(3) ) , (138 + 1.645 (26.90)/(√(3) ) )`

( 138 - 25.54 , 138 + 25.54 )

(112.46 , 163.54)