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 (𝑥̄) and the sample standard deviation (𝑠) from the given data.
The sample mean (𝑥̄) can be calculated by summing up all the values and dividing by the total number of values.
In this case, we have:
𝑥̄ = (21*1 + 22*2 + 23*8 + 24*5 + 25*8) / (1 + 2 + 8 + 5 + 8) = 23.116
The sample standard deviation (𝑠) can be calculated using the formula:
𝑠 = sqrt[ (Σ(𝑋-𝑥̄)^2) / (𝑛-1) ]
where 𝑋 is each individual value in the sample, 𝑥̄ is the sample mean, and 𝑛 is the number of values in the sample.
To calculate 𝑠, we need to calculate the sum of squares of deviations from the mean (Σ(𝑋-𝑥̄)^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 𝑠.
Step 2: Calculate the standard error (𝑠𝑒) using the formula:
𝑠𝑒 = 𝑠 / sqrt(𝑛)
where 𝑠 is the sample standard deviation and 𝑛 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 (𝑀𝐸) using the formula:
𝑀𝐸 = 𝑍 * 𝑠𝑒
where 𝑍 is the critical value for the desired confidence level and 𝑠𝑒 is the standard error.
Step 5: Construct the confidence interval.
The confidence interval can be calculated by using the formula:
𝑥̄ ± 𝑀𝐸
Substituting the values, we have:
Confidence interval = 23.116 ± 𝑀𝐸
Finally, calculate the lower bound and upper bound of the confidence interval by subtracting and adding the margin of error:
Lower bound = 23.116 - 𝑀𝐸
Upper bound = 23.116 + 𝑀𝐸
This way, you can construct a 95% confidence interval for the population mean airfare based on the given data.