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
To construct a 95% confidence interval for the population mean airfare, we can use the following steps:
Step 1: Calculate the sample mean (π₯Μ) and the sample standard deviation (π ) from the given data.
The sample mean (π₯Μ) can be calculated by summing up all the values and dividing by the total number of values.
In this case, we have:
π₯Μ = (21*1 + 22*2 + 23*8 + 24*5 + 25*8) / (1 + 2 + 8 + 5 + 8) = 23.116
The sample standard deviation (π ) can be calculated using the formula:
π = sqrt[ (Ξ£(π-π₯Μ)^2) / (π-1) ]
where π is each individual value in the sample, π₯Μ is the sample mean, and π is the number of values in the sample.
To calculate π , we need to calculate the sum of squares of deviations from the mean (Ξ£(π-π₯Μ)^2):
[(21-23.116)^2]*1 + [(22-23.116)^2]*2 + [(23-23.116)^2]*8 + [(24-23.116)^2]*5 + [(25-23.116)^2]*8
After calculating this sum, we will divide it by 30-1=29 and take the square root to get π .
Step 2: Calculate the standard error (π π) using the formula:
π π = π / sqrt(π)
where π is the sample standard deviation and π is the number of values in the sample.
Step 3: Determine the critical value for a 95% confidence level.
The critical value depends on the sample size and the desired confidence level. In this case, we have a sample size of 30, and since we want a 95% confidence level, the critical value (Z) is approximately 1.96 (from a standard normal distribution table).
Step 4: Calculate the margin of error (ππΈ) using the formula:
ππΈ = π * π π
where π is the critical value for the desired confidence level and π π is the standard error.
Step 5: Construct the confidence interval.
The confidence interval can be calculated by using the formula:
π₯Μ Β± ππΈ
Substituting the values, we have:
Confidence interval = 23.116 Β± ππΈ
Finally, calculate the lower bound and upper bound of the confidence interval by subtracting and adding the margin of error:
Lower bound = 23.116 - ππΈ
Upper bound = 23.116 + ππΈ
This way, you can construct a 95% confidence interval for the population mean airfare based on the given data.