Construct a 95% confidence interval for the population mean. 5. Airfare A random sample of airfare prices (in dollars) for a one-way ticket airfare between two cities MN. Key: 21|1=211
21 | 1 2
22 | 1 1 1 2 3 5 7 8
23 | 2 3 4 7 7 8 9 9
24 | 2 3 4 8 8
25 | 1 2 6 6 6 6 8 9
95% = mean ± 2 SD
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
To construct a 95% confidence interval for the population mean, we need to find the sample mean, sample standard deviation, sample size, and the critical value from the t-distribution.
From the provided data:
Sample Mean (x̄) = (21*1 + 22*2 + 23*8 + 24*5 + 25*8) / (1 + 2 + 8 + 5 + 8) = 575 / 24 = 23.96
To find the sample standard deviation, we first need to calculate the sum of squares of the differences between each observation and the sample mean:
Sum of Squares = (1-23.96)^2 + (2-23.96)^2 + (8-23.96)^2 + (5-23.96)^2 + (8-23.96)^2 + (1-23.96)^2 + (1-23.96)^2 + (1-23.96)^2 + (2-23.96)^2 + (3-23.96)^2 + (5-23.96)^2 + (7-23.96)^2 + (8-23.96)^2 + (2-23.96)^2 + (3-23.96)^2 + (4-23.96)^2 + (7-23.96)^2 + (7-23.96)^2 + (8-23.96)^2 + (9-23.96)^2 + (9-23.96)^2 + (2-23.96)^2 + (3-23.96)^2 + (4-23.96)^2 + (8-23.96)^2 + (8-23.96)^2 + (1-23.96)^2 + (2-23.96)^2 + (6-23.96)^2 + (6-23.96)^2 + (6-23.96)^2 + (6-23.96)^2 + (8-23.96)^2 + (9-23.96)^2
Sum of Squares = 1139.44
Sample Standard Deviation (s) = sqrt(Sum of Squares / (n - 1)) = sqrt(1139.44 / 23) = sqrt(49.549) = 7.03
Sample Size (n) = 24
The critical value from the t-distribution can be obtained using the degrees of freedom (n - 1) and the desired confidence level (95%). Since the sample size is 24, the degrees of freedom is 23.
To find the critical value, you can use a t-table or a calculator. For a 95% confidence level with 23 degrees of freedom, the critical value is approximately 2.069.
Now, we can calculate the margin of error and construct the confidence interval.
Margin of Error = Critical Value * (Sample Standard Deviation / sqrt(Sample Size))
= 2.069 * (7.03 / sqrt(24))
= 2.069 * (7.03 / 4.899)
= 2.969
Confidence Interval = Sample Mean ± Margin of Error
= 23.96 ± 2.969
Therefore, the 95% confidence interval for the population mean airfare prices (in dollars) for a one-way ticket between the two cities in MN is approximately (20.99, 26.93).