Annual cost of College. The increasing annual cost...

Annual cost of College. The increasing annual cost (including tuition, room, board, books, and fees) to attend college has been widely discussed in many publications including Money magazine. The following random samples show the annual cost of attending private and public colleges. Data are in thousands of dollars. $$ \begin{aligned} &\text { Private Colleges }\\ &\begin{array}{lllll} 52.8 & 43.2 & 45.0 & 33.3 & 44.0 \\ 30.6 & 45.8 & 37.8 & 50.5 & 42.0 \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Public Colleges }\\ &\begin{array}{llllll} 20.3 & 22.0 & 28.2 & 15.6 & 24.1 & 28.5 \\ 22.8 & 25.8 & 18.5 & 25.6 & 14.4 & 21.8 \end{array} \end{aligned} $$ a. Compute the sample mean and sample standard deviation for private and public colleges. b. What is the point estimate of the difference between the two population means? Interpret this value in terms of the annual cost of attending private and public colleges. c. Develop a $95 \%$ confidence interval of the difference between the mean annual cost of attending private and public colleges.

Annual cost of College. The increasing annual cost (including tuition, room, board, books, and fees) to attend college has been widely discussed in many publications including Money magazine. The following random samples show the annual cost of attending private and public colleges. Data are in thousands of dollars.
$$
\begin{aligned}
&\text { Private Colleges }\\
&\begin{array}{lllll}
52.8 & 43.2 & 45.0 & 33.3 & 44.0 \\
30.6 & 45.8 & 37.8 & 50.5 & 42.0
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\text { Public Colleges }\\
&\begin{array}{llllll}
20.3 & 22.0 & 28.2 & 15.6 & 24.1 & 28.5 \\
22.8 & 25.8 & 18.5 & 25.6 & 14.4 & 21.8
\end{array}
\end{aligned}
$$
a. Compute the sample mean and sample standard deviation for private and public colleges.
b. What is the point estimate of the difference between the two population means? Interpret this value in terms of the annual cost of attending private and public colleges.
c. Develop a $95 \%$ confidence interval of the difference between the mean annual cost of attending private and public colleges.

Key Concepts

Confidence Interval for the Difference Between Two Means

A confidence interval for the difference between two means provides a range of values that, with a certain level of confidence (such as 95%), is believed to contain the true difference between the population means. This interval is constructed using the point estimate plus or minus a margin of error, which accounts for the variability in the data and the sample sizes. It gives a measure of the precision of the point estimate and helps assess the statistical significance of the observed difference.

Point Estimate for the Difference Between Two Population Means

A point estimate for the difference between two population means is obtained by subtracting one sample mean from the other. This difference serves as an estimate of how much the average values differ between the two populations. It is a central tool in comparative studies, such as comparing the annual costs between private and public colleges, providing a single best estimate of the true difference in population parameters.

Independent Samples

Independent samples refer to two groups of observations that are collected in such a way that the selection of one sample does not influence or inform the selection of the other sample. In the context of comparing the costs of attending private and public colleges, each group is independent of the other, which justifies the use of statistical methods designed for comparing separate populations.

Sample Mean

The sample mean is the arithmetic average of a set of observations and serves as an estimate of the true population mean. It is computed by summing all the observations in the sample and then dividing by the total number of observations. In statistical inference, the sample mean is used as a basic point estimator that provides insight into the central tendency of the population from which the sample was drawn.

Sample Standard Deviation

The sample standard deviation measures the variability or dispersion of the sample data around the sample mean. It is calculated by first computing the squared differences between each observation and the sample mean, summing these squared differences, dividing by the degrees of freedom (typically the sample size minus one), and finally taking the square root of the result. This statistic provides insight into the spread of the data and is essential when comparing variability between different samples.

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