Expedia would like to test the hypothesis that the average round trip airfare between Orlando and Salt Lake is the same for a flight originating in Orlando when compared to a flight originating in Salt Lake.
The Correct Answer and Explanation is:
The correct answer is (-$25.14, $33.14).
Here is a step-by-step explanation of how to arrive at this answer:
This problem requires the calculation of a 95% confidence interval for the difference between two independent population means. Since the population standard deviations are unknown, we use a two-sample t-interval. The problem specifies using 18 degrees of freedom, which corresponds to the formula for a pooled t-interval, as the degrees of freedom are calculated by (n₁ + n₂ – 2).
1. Identify the Sample Data:
Let Population 1 be flights from Orlando and Population 2 be flights from Salt Lake.
- Orlando (Sample 1):
- Sample Mean (x̄₁): $171
- Sample Standard Deviation (s₁): $30
- Sample Size (n₁): 10
- Salt Lake (Sample 2):
- Sample Mean (x̄₂): $167
- Sample Standard Deviation (s₂): $32
- Sample Size (n₂): 10
2. Calculate the Difference in Sample Means:
The difference between the sample means is:
x̄₁ – x̄₂ = $171 – $167 = $4
3. Calculate the Pooled Variance (s_p²):
The pooled variance combines the variances of both samples:
s_p² = [ (n₁-1)s₁² + (n₂-1)s₂² ] / (n₁ + n₂ – 2)
s_p² = [ (10-1)(30)² + (10-1)(32)² ] / (10 + 10 – 2)
s_p² = [ 9(900) + 9(1024) ] / 18
s_p² = [ 8100 + 9216 ] / 18
s_p² = 17316 / 18 = 962
4. Calculate the Standard Error (SE):
The standard error of the difference between the means is:
SE = √[ s_p² * (1/n₁ + 1/n₂) ]
SE = √[ 962 * (1/10 + 1/10) ]
SE = √[ 962 * (0.2) ]
SE = √192.4 ≈ 13.871
5. Determine the Critical t-value (t):*
For a 95% confidence interval with 18 degrees of freedom, the critical t-value (t*) is 2.101.
6. Calculate the Margin of Error (ME):
The margin of error is the critical t-value multiplied by the standard error:
ME = t* * SE
ME = 2.101 * 13.871 ≈ 29.143
7. Construct the Confidence Interval:
The confidence interval is calculated as (Difference in Means) ± (Margin of Error):
Interval = ($4 – $29.143, $4 +
29.143)Interval=(−29.143)Interval=(−25.143, $33.143)
Rounding to two decimal places, the 95% confidence interval for the difference in population means is (-$25.14, $33.14).
