In Exercise 3, after deleting an outlying value of -27, the...

In Exercise $3,$ after deleting an outlying value of $-27,$ the mean difference in fuel efficiencies for the 632 vehicles was $7.37 \mathrm{mpg}$ with a standard deviation of $2.52 \mathrm{mpg} .$ Find a $95 \%$ confidence interval for this difference and interpret it in context.

Key Concepts

Confidence Interval

A confidence interval is a range constructed from sample data that is likely to contain the true value of an unknown population parameter. It reflects the reliability of an estimate by providing a lower and upper bound, based on the chosen confidence level (such as 95%), and accounts for the variability inherent in sampling.

Standard Error

The standard error quantifies the variability of a sample statistic, such as the sample mean, by reflecting how much it is expected to vary from the true population parameter. It is calculated by dividing the sample standard deviation by the square root of the sample size, providing a measure of the precision of the estimate.

t-Distribution

The t-distribution is used instead of the normal distribution when the sample size is small or the population standard deviation is unknown. It compensates for additional uncertainty in the estimation process by having heavier tails, and its exact shape depends on the degrees of freedom associated with the sample.

Margin of Error

The margin of error expresses the extent of uncertainty around a sample statistic, calculated by multiplying the critical value from the t-distribution (or normal distribution for large samples) by the standard error. It defines how far above and below the sample statistic the true population parameter is expected to lie, within the specified confidence level.

Interpretation of Confidence Interval

Interpreting a confidence interval means understanding that if the same sampling method were repeated many times, a certain percentage (e.g., 95%) of the computed intervals would contain the true population parameter. It communicates the reliability of the estimation process without implying that there is a direct probability for the parameter to lie within any one computed interval.

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