Estimate the mean of the following set of data to one decimal place: * $\newline$ \begin{tabular}{|c|c|} $\newline$ \hline Number of Words & Frequency \\ $\newline$ \hline $1-7$ & $15$ \\ $\newline$ \hline $8-14$ & $20$ \\ $\newline$ $15-21$ & $45$ \\ $\newline$ $22-28$ & $47$ \\ $\newline$ $29-35$ & $23$ \\ $\newline$ $36-42$ & $10$ \\ $\newline$ \hline $\newline$ \end{tabular}

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Q. Estimate the mean of the following set of data to one decimal place: *

\newline

\begin{tabular}{|c|c|}

\newline

\hline Number of Words & Frequency \\

\newline

\hline

1-7

15

\newline

\hline

8-14

20

\newline

15-21

45

\newline

22-28

47

\newline

29-35

23

\newline

36-42

10

\newline

\hline

\newline

\end{tabular}

Find Midpoints: First, we need to find the midpoint of each class interval. This is done by adding the lower and upper bounds and dividing by $2$ . $\newline$ For $1-7$ , the midpoint is $(1+7)/2 = 4$ . $\newline$ For $8-14$ , the midpoint is $(8+14)/2 = 11$ . $\newline$ For $15-21$ , the midpoint is $(15+21)/2 = 18$ . $\newline$ For $22-28$ , the midpoint is $(22+28)/2 = 25$ . $\newline$ For $29-35$ , the midpoint is $1-7$ $0$ . $\newline$ For $1-7$ $1$ , the midpoint is $1-7$ $2$ .
Calculate Weighted Midpoints: Next, we multiply each midpoint by its frequency to find the ＂weighted＂ midpoint. $\newline$ For $1$ $-7$ , it's $4 \times 15 = 60$ . $\newline$ For $8$ $-14$ , it's $11 \times 20 = 220$ . $\newline$ For $15$ $-21$ , it's $18 \times 45 = 810$ . $\newline$ For $22$ $-28$ , it's $25 \times 47 = 1175$ . $\newline$ For $29$ $-35$ , it's $32 \times 23 = 736$ . $\newline$ For $36$ $-42$ , it's $39 \times 10 = 390$ .
Add Weighted Midpoints: Now, we add up all the ＂weighted＂ midpoints to get the total. So, the total is $60 + 220 + 810 + 1175 + 736 + 390 = 3391$ .
Find Total Frequencies: Then, we need to find the sum of all frequencies to get the total number of data points. $\newline$ The sum is $15 + 20 + 45 + 47 + 23 + 10 = 160$ .
Calculate Mean: Finally, we divide the total of the ＂weighted＂ midpoints by the sum of the frequencies to find the mean. $\newline$ The mean is $\frac{3391}{160} = 21.19375$ .
Round Mean: Round the mean to one decimal place. $\newline$ The mean rounded to one decimal place is $21.2$ .