One gallon of gasoline is put into each of an SRS of 30 test...

One gallon of gasoline is put into each of an SRS of 30 test autos, and the resulting mileage figures are tabulated with $\bar{x}$ $=28.5$ and $s=1.2 .$ Determine a $95 \%$ confidence interval estimate of the mean mileage of all comparable autos. (A) $28.5 \pm 2.045(1.2)$ (B) $28.5 \pm 2.045\left(\frac{1.2}{\sqrt{29}}\right)$ (C) $28.5 \pm 2.045\left(\frac{1.2}{\sqrt{29}}\right)$ (D) $28.5 \pm 1.96\left(\frac{1.2}{\sqrt{29}}\right)$ (E) $28.5 \pm 1.96\left(\frac{1.2}{\sqrt{29}}\right)$

One gallon of gasoline is put into each of an SRS of 30 test autos, and the resulting mileage figures are tabulated with $\bar{x}$ $=28.5$ and $s=1.2 .$ Determine a $95 \%$ confidence interval estimate of the mean mileage of all comparable autos.
(A) $28.5 \pm 2.045(1.2)$
(B) $28.5 \pm 2.045\left(\frac{1.2}{\sqrt{29}}\right)$
(C) $28.5 \pm 2.045\left(\frac{1.2}{\sqrt{29}}\right)$
(D) $28.5 \pm 1.96\left(\frac{1.2}{\sqrt{29}}\right)$
(E) $28.5 \pm 1.96\left(\frac{1.2}{\sqrt{29}}\right)$

Key Concepts

Confidence Interval

A confidence interval provides a range of plausible values for an unknown population parameter, based on sample data. It is constructed so that, under repeated sampling, the interval would contain the true parameter a specified percentage of the time, such as 95% in many applications.

Student's t-Distribution

When the population standard deviation is unknown and the sample size is relatively small, the sampling distribution of the sample mean is better approximated by a t-distribution rather than a normal distribution. The t-distribution accounts for extra variability due to estimating the population standard deviation from the sample.

Degrees of Freedom

Degrees of freedom in the context of a t-distribution refer to the number of independent values that can vary after accounting for the estimated parameters. For a sample mean computed from n observations, the degrees of freedom are usually n - 1, which influences the critical values used in constructing a confidence interval.

Standard Error

The standard error quantifies the variability in the estimate of the population mean from one sample to another and is calculated as the sample standard deviation divided by the square root of the sample size. It plays a crucial role in determining the width of the confidence interval.

Margin of Error

The margin of error represents the extent of the confidence interval on either side of the sample estimate and is calculated as the product of the t (or z) critical value and the standard error. It reflects the level of uncertainty associated with the estimate of the population parameter.

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