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For the following set of data, find the percentage of data within 2 population standard deviations of the mean, to the nearest percent.

Data
Frequency

6
2

9
7

10
10

11
14

15
18

16
15

18
9

19
8

20
1

For the following set of data, find the percentage of data within $2$ population standard deviations of the mean, to the nearest percent. $\newline$ \begin{tabular}{|c|c|} $\newline$ \hline Data & Frequency \\ $\newline$ \hline $6$ & $2$ \\ $\newline$ \hline $9$ & $7$ \\ $\newline$ \hline $10$ & $10$ \\ $\newline$ \hline $11$ & $14$ \\ $\newline$ \hline $15$ & $18$ \\ $\newline$ \hline $16$ & $15$ \\ $\newline$ \hline $18$ & $9$ \\ $\newline$ \hline $19$ & $8$ \\ $\newline$ \hline $20$ & $1$ \\ $\newline$ \hline $\newline$ \end{tabular}

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Interpret measures of center and variability

Full solution

Q. For the following set of data, find the percentage of data within

2

population standard deviations of the mean, to the nearest percent.

\newline

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

\newline

\hline Data & Frequency \\

\newline

\hline

6

&

2

\\

\newline

\hline

9

&

7

\\

\newline

\hline

10

&

10

\\

\newline

\hline

11

&

14

\\

\newline

\hline

15

&

18

\\

\newline

\hline

16

&

15

\\

\newline

\hline

18

&

9

\\

\newline

\hline

19

&

8

\\

\newline

\hline

20

&

1

\\

\newline

\hline

\newline

\end{tabular}

Calculate Mean: Calculate the mean of the data set. $\newline$ Mean = $(\text{Sum of all values}) / (\text{Total number of values})$ $\newline$ = $(6\times2 + 9\times7 + 10\times10 + 11\times14 + 15\times18 + 16\times15 + 18\times9 + 19\times8 + 20\times1) / (2 + 7 + 10 + 14 + 18 + 15 + 9 + 8 + 1)$ $\newline$ = $(12 + 63 + 100 + 154 + 270 + 240 + 162 + 152 + 20) / 84$ $\newline$ = $1073 / 84$ $\newline$ = $12.77$ (rounded to two decimal places)
Calculate Standard Deviation: Calculate the standard deviation of the data set. $\newline$ First, find the variance by using the formula: Variance = $\frac{\text{Sum of } (\text{value} - \text{mean})^2 \times \text{frequency}}{\text{Total number of values}}$ $\newline$ Variance = $\frac{(6-12.77)^2\times2 + (9-12.77)^2\times7 + (10-12.77)^2\times10 + (11-12.77)^2\times14 + (15-12.77)^2\times18 + (16-12.77)^2\times15 + (18-12.77)^2\times9 + (19-12.77)^2\times8 + (20-12.77)^2\times1}{84}$ $\newline$ = $\frac{(-6.77)^2\times2 + (-3.77)^2\times7 + (-2.77)^2\times10 + (-1.77)^2\times14 + (2.23)^2\times18 + (3.23)^2\times15 + (5.23)^2\times9 + (6.23)^2\times8 + (7.23)^2\times1}{84}$ $\newline$ = $\frac{91.5729\times2 + 14.2129\times7 + 7.6729\times10 + 3.1329\times14 + 4.9729\times18 + 10.4329\times15 + 27.3529\times9 + 38.8129\times8 + 52.2729\times1}{84}$ $\newline$ = $\frac{183.1458 + 99.4903 + 76.729 + 43.8606 + 89.5112 + 156.4935 + 246.1761 + 310.5032 + 52.2729}{84}$ $\newline$ = $\frac{1258.1826}{84}$ $\newline$ = $14$ . $9786$ (rounded to four decimal places) $\newline$ Standard deviation = $\sqrt{\text{Variance}}$ $\newline$ = $\sqrt{14.9786}$ $\newline$ = $3$ . $87$ (rounded to two decimal places)
Determine Range: Determine the range within $2$ standard deviations of the mean. $\newline$ Lower bound = Mean - $2$ $\times$ Standard deviation $\newline$ $= 12.77 - 2 \times 3.87$ $\newline$ $= 12.77 - 7.74$ $\newline$ $= 5.03$ (rounded to two decimal places) $\newline$ Upper bound = Mean + $2$ $\times$ Standard deviation $\newline$ $= 12.77 + 2 \times 3.87$ $\newline$ $= 12.77 + 7.74$ $\newline$ $= 20.51$ (rounded to two decimal places)
Count Values in Range: Count the number of values within the range of $5.03$ to $20.51$ . Values within range: $6$ , $9$ , $10$ , $11$ , $15$ , $16$ , $18$ , $19$ ( $20.51$ $0$ is slightly outside the range) Frequency within range: $20.51$ $1$
Calculate Percentage: Calculate the percentage of values within $2$ standard deviations. $\newline$ Percentage = $(\frac{\text{Frequency within range}}{\text{Total number of values}}) \times 100$ $\newline$ = $(\frac{83}{84}) \times 100$ $\newline$ = $98$ . $81$ \% (rounded to nearest percent)

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