Question

The city planners wanted to conduct a study of the speeds of vehicles being driven on a certain road. They tracked the speeds of 23 randomly selected vehicles and found that the mean speed was 31 mph and standard deviation was 4.25 mph.

1. a) Determine a 90% confidence interval for the mean speed of all vehicles being driven on that

   road. Explain the underlying assumptions and the meaning of this confidence interval.

2. b) If the planners wanted to determine mean speed with a margin-of-error of 0.5 miles, what

   sample size would be required at 90% confidence level?

3. c) If the planners had tested a "large" sample (n ≥ 30), would the 90% confidence interval have

   been narrower or wider than in part (a). Explain.

Answer 1

a)

90% Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.1 /2, 23- 1 ) = 1.717
31 ± t(0.1/2, 23 -1) * 4.25/√(23)
Lower Limit = 31 - t(0.1/2, 23 -1) 4.25/√(23)
Lower Limit = 29.48
Upper Limit = 31 + t(0.1/2, 23 -1) 4.25/√(23)
Upper Limit = 32.52
90% Confidence interval is ( 29.48 , 32.52 )

b)

Marin of error = ( t * S / E)²

= ( 1.717 * 4.25 / 0.5)²

= 213

c)

For larger sample size, margin of error would be smaller.

90% confidence interval would have been narrower.