Question

A CI is desired for the true average stray-load loss μ (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with σ = 3.3. (Round your answers to two decimal places.)

(a) Compute a 95% CI for μ when n = 25 and x = 54.7.

,

watts

(b) Compute a 95% CI for μ when n = 100 and x = 54.7.

,

watts

(c) Compute a 99% CI for μ when n = 100 and x = 54.7.

,

watts

(d) Compute an 82% CI for μ when n = 100 and x = 54.7.

,

watts

(e) How large must n be if the width of the 99% interval for μ is to be 1.0? (Round your answer up to the nearest whole number.)
n =

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

Part a)

Confidence Interval :-
X̅ ± Z( α /2) σ / √ ( n )
Z(α/2) = Z (0.05 /2) = 1.96
54.7 ± Z (0.05/2 ) * 3.3/√(25)
Lower Limit = 54.7 - Z(0.05/2) 3.3/√(25)
Lower Limit = 53.41
Upper Limit = 54.7 + Z(0.05/2) 3.3/√(25)
Upper Limit = 55.99
95% Confidence interval is ( 53.41 , 55.99 )

Part b)

Confidence Interval :-
X̅ ± Z( α /2) σ / √ ( n )
Z(α/2) = Z (0.05 /2) = 1.96
54.7 ± Z (0.05/2 ) * 3.3/√(100)
Lower Limit = 54.7 - Z(0.05/2) 3.3/√(100)
Lower Limit = 54.05
Upper Limit = 54.7 + Z(0.05/2) 3.3/√(100)
Upper Limit = 55.35
95% Confidence interval is ( 54.05 , 55.35 )

Part c)

Confidence Interval :-
X̅ ± Z( α /2) σ / √ ( n )
Z(α/2) = Z (0.01 /2) = 2.576
54.7 ± Z (0.01/2 ) * 3.3/√(100)
Lower Limit = 54.7 - Z(0.01/2) 3.3/√(100)
Lower Limit = 53.85
Upper Limit = 54.7 + Z(0.01/2) 3.3/√(100)
Upper Limit = 55.55
99% Confidence interval is ( 53.85 , 55.55 )

Part d)

Confidence Interval :-
X̅ ± Z( α /2) σ / √ ( n )
Z(α/2) = Z (0.18 /2) = 1.341
54.7 ± Z (0.18/2 ) * 3.3/√(100)
Lower Limit = 54.7 - Z(0.18/2) 3.3/√(100)
Lower Limit = 54.26
Upper Limit = 54.7 + Z(0.18/2) 3.3/√(100)
Upper Limit = 55.14
82% Confidence interval is ( 54.26 , 55.14 )

Part e)

Sample size can be calculated by below formula
n = (( Z(α/2) * σ) / e )2
n = (( Z(0.01/2) * 3.3 ) / 1 )2
Critical value Z(α/2) = Z(0.01/2) = 2.5758
n = (( 2.5758 * 3.3 ) / 1 )2
n = 73
Required sample size at 99% confidence is 73.