For a random sample of 25 students from a very large...

For a random sample of 25 students from a very large university, the accompanying table shows the amount of time (in hours) spent studying for final exams. $$ \begin{array}{lccccc} \hline \text { Study time } & 0<4 & 4<8 & 8<12 & 12<16 & 16<20 \\ \hline \begin{array}{l} \text { Number of } \\ \text { students } \end{array} & 3 & 7 & 8 & 5 & 2 \\ \hline \end{array} $$ a. Estimate the sample mean study time. b. Estimate the sample standard deviation.

For a random sample of 25 students from a very large university, the accompanying table shows the amount of time (in hours) spent studying for final exams.
$$
\begin{array}{lccccc}
\hline \text { Study time } & 0<4 & 4<8 & 8<12 & 12<16 & 16<20 \\
\hline \begin{array}{l}
\text { Number of } \\
\text { students }
\end{array} & 3 & 7 & 8 & 5 & 2 \\
\hline
\end{array}
$$
a. Estimate the sample mean study time.
b. Estimate the sample standard deviation.

Key Concepts

Grouped Data

Grouped data refers to data that has been organized into intervals or groups, where individual data points are not available but their frequencies are. This format is common for large datasets and facilitates easier data analysis, though it necessitates estimation techniques since exact values are not provided.

Midpoint Calculation

When data is grouped into intervals, the midpoint of each interval is typically used as a representative value for all observations within that interval. The midpoint is calculated by averaging the lower and upper bounds of the interval, and it serves as an estimate for the data points in the group.

Weighted Mean

The weighted mean is an estimation of the average value for grouped data, calculated by multiplying each interval's midpoint by its frequency, summing these products, and then dividing by the total number of observations. This approach accounts for the differing frequencies across the intervals.

Weighted Standard Deviation

The weighted standard deviation is an estimate of the dispersion or spread of the data in a grouped frequency distribution. It is calculated by determining the squared deviation of each interval's midpoint from the weighted mean, multiplying by the frequency for that interval, summing these products, and then dividing by the number of observations minus one before taking the square root. This method adjusts for the grouping and provides a measure of variability.

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