A car dealership wants to estimate the average mileage of a new model of car. They take a random sample of 15 cars and record their mileage. The sample mean mileage is 30 miles per gallon and the sample standard deviation is 2 miles per gallon. Calculate a 90% confidence interval for the population mean mileage.
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Step-by-step explanation:
To calculate the 90% confidence interval for the population mean mileage, we will use the t-distribution, as the population standard deviation is unknown.
Here's the information given:
- Sample size (n) = 15
- Sample mean (x̄) = 30 miles per gallon
- Sample standard deviation (s) = 2 miles per gallon
- Confidence level = 90%
First, let's find the degrees of freedom:
Degrees of freedom (df) = n - 1 = 15 - 1 = 14
Now, we need to find the t-score for a 90% confidence interval and 14 degrees of freedom. Using a t-table or calculator, the corresponding t-score is approximately 1.761.
Next, we'll calculate the margin of error (ME). The formula for ME is:
ME = t-score * (s / sqrt(n))
ME = 1.761 * (2 / sqrt(15))
ME ≈ 0.908
Now, we can calculate the confidence interval using the sample mean and the margin of error:
Lower limit = x̄ - ME = 30 - 0.908 ≈ 29.092
Upper limit = x̄ + ME = 30 + 0.908 ≈ 30.908
Thus, the 90% confidence interval for the population mean mileage is approximately (29.092, 30.908) miles per gallon.