Fifteen smart cars were randomly selected and the highway mileage of each was noted. The analysis yielded a mean of 47 miles per gallon and a standard deviation of 5 miles per galon. Which of the following would represent a 90% confidence interval for the average highway mileage of all SmartCars?

The correct answer is:

47-/+1.761(5/√15)

but my question is: how do you get 1.761?

Thanks

On a normal distributed cureve, mean=0, the standard deviation to have for a 90% confidence interval for the mean µ is t0.05,14 = 1.761 . The "cumulative probability" to the left of 1.76 is 0.95, and the probability to the right of 1.76 is 0.05.

The value 1.761 corresponds to the critical value needed for a 90% confidence interval with a sample size of 15.

To find the critical value, we can use the t-distribution table or a statistical calculator.

For a 90% confidence interval with a sample size of 15, we would use the t-distribution with 15-1=14 degrees of freedom.

By looking up the critical value for a 90% confidence level and 14 degrees of freedom, we find the value of approximately 1.761.

This value accounts for the desired level of confidence and the variability of the data, helping to determine the appropriate range for the confidence interval.

To understand how to get the value of 1.761, we need to know that it corresponds to a specific critical value from the t-distribution table.

In this context, a t-distribution is used because the population standard deviation is unknown, and we are estimating it using the sample's standard deviation. The critical value from the t-distribution is used to determine the amount of variability or uncertainty in the estimate of the population mean.

The value of 1.761 represents the critical value for a 90% confidence level, with 14 degrees of freedom (n-1, where n is the sample size). It is obtained from the t-distribution table or can be calculated using software or calculators specialized in statistical analyses.

The critical value depends on the desired confidence level and the sample size. In this case, the 90% confidence level is chosen, meaning that we want to be 90% confident that the true population mean lies within the calculated confidence interval.

By multiplying the critical value with the standard error of the mean, we can determine the margin of error that should be added or subtracted to the sample mean to construct the confidence interval.

In the case of the provided answer, the formula for the confidence interval is as follows:

Confidence Interval = Sample Mean ± (Critical Value) × (Standard Deviation / sqrt(sample size))

Substituting the given values:

Confidence Interval = 47 ± (1.761) × (5 / sqrt(15))

Hence, the resulting confidence interval is 47 ± 1.761(5/sqrt(15)), which gives you the range within which the true population mean is estimated to lie with 90% confidence.