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A study on the effects of exercise on the menstrual cycle provides the following ages (years) of menarche (the beginning of menstruation) for 96 female swimmers who began training prior to menarche.

15.0
17.1
14.6
15.2
14.9
14.4
14.7
15.3
13.6
15.1
15.3
14.7

16.2
15.9
13.8
15.0
15.4
14.9
14.2
16.5
13.2
16.8
13.9
16.1

15.4
14.6
15.2
14.8
13.7
16.3
15.1
14.5
16.4
13.6
14.8
15.5

13.9
15.9
16.0
14.6
14.0
15.1
14.8
15.0
14.8
15.3
15.7
14.3

13.9
15.6
15.4
14.6
15.2
14.8
13.7
16.3
15.1
14.5
13.6
15.1

16.2
15.9
13.8
15.0
15.4
14.9
16.2
15.9
13.8
15.0
15.4
14.9

14.2
16.5
13.4
16.5
14.8
15.1
14.9
13.7
16.2
15.8
15.4
14.7

14.3
15.2
14.6
13.7
14.9
15.8
15.1
14.6
13.8
16.0
15.0
14.6

Construct the grouped frequency distribution for the above given data using 'inclusive class intervals' and 'exclusive class intervals. Also report how many numbers of classes are required for the given data.
Also construct Histogram and frequency curve.

$2$ . A study on the effects of exercise on the menstrual cycle provides the following ages (years) of menarche (the beginning of menstruation) for $96$ female swimmers who began training prior to menarche. $\newline$ \begin{tabular}{llllllllllll} $\newline$ $15$ . $0$ & $17$ . $1$ & $14$ . $6$ & $15$ . $2$ & $14$ . $9$ & $14$ . $4$ & $14$ . $7$ & $15$ . $3$ & $13$ . $6$ & $15$ . $1$ & $15$ . $3$ & $14$ . $7$ \\ $\newline$ $16$ . $2$ & $15$ . $9$ & $13$ . $8$ & $15$ . $0$ & $15$ . $4$ & $14$ . $9$ & $14$ . $2$ & $16$ . $5$ & $13$ . $2$ & $16$ . $8$ & $13$ . $9$ & $16$ . $1$ \\ $\newline$ $15$ . $4$ & $14$ . $6$ & $15$ . $2$ & $14$ . $8$ & $13$ . $7$ & $16$ . $3$ & $15$ . $1$ & $14$ . $5$ & $16$ . $4$ & $13$ . $6$ & $14$ . $8$ & $15$ . $5$ \\ $\newline$ $13$ . $9$ & $15$ . $9$ & $16$ . $0$ & $14$ . $6$ & $14$ . $0$ & $15$ . $1$ & $14$ . $8$ & $15$ . $0$ & $14$ . $8$ & $15$ . $3$ & $15$ . $7$ & $14$ . $3$ $\newline$ \end{tabular} $\newline$ \begin{tabular}{llllllllllll} $\newline$ $13$ . $9$ & $15$ . $6$ & $15$ . $4$ & $14$ . $6$ & $15$ . $2$ & $14$ . $8$ & $13$ . $7$ & $16$ . $3$ & $15$ . $1$ & $14$ . $5$ & $13$ . $6$ & $15$ . $1$ \\ $\newline$ $16$ . $2$ & $15$ . $9$ & $13$ . $8$ & $15$ . $0$ & $15$ . $4$ & $14$ . $9$ & $16$ . $2$ & $15$ . $9$ & $13$ . $8$ & $15$ . $0$ & $15$ . $4$ & $14$ . $9$ \\ $\newline$ $14$ . $2$ & $16$ . $5$ & $13$ . $4$ & $16$ . $5$ & $14$ . $8$ & $15$ . $1$ & $14$ . $9$ & $13$ . $7$ & $16$ . $2$ & $15$ . $8$ & $15$ . $4$ & $14$ . $7$ \\ $\newline$ $14$ . $3$ & $15$ . $2$ & $14$ . $6$ & $13$ . $7$ & $14$ . $9$ & $15$ . $8$ & $15$ . $1$ & $14$ . $6$ & $13$ . $8$ & $16$ . $0$ & $15$ . $0$ & $14$ . $6$ $\newline$ \end{tabular} $\newline$ $1$ . Construct the grouped frequency distribution for the above given data using 'inclusive class intervals' and 'exclusive class intervals. Also report how many numbers of classes are required for the given data. $\newline$ $2$ . Also construct Histogram and frequency curve.

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Full solution

Q.

2

. A study on the effects of exercise on the menstrual cycle provides the following ages (years) of menarche (the beginning of menstruation) for

96

female swimmers who began training prior to menarche.

\newline

\begin{tabular}{llllllllllll}

\newline

15

.

0

&

17

.

1

&

14

.

6

&

15

.

2

&

14

.

9

&

14

.

4

&

14

.

7

&

15

.

3

&

13

.

6

&

15

.

1

&

15

.

3

&

14

.

7

\\

\newline

16

.

2

&

15

.

9

&

13

.

8

&

15

.

0

&

15

.

4

&

14

.

9

&

14

.

2

&

16

.

5

&

13

.

2

&

16

.

8

&

13

.

9

&

16

.

1

\\

\newline

15

.

4

&

14

.

6

&

15

.

2

&

14

.

8

&

13

.

7

&

16

.

3

&

15

.

1

&

14

.

5

&

16

.

4

&

13

.

6

&

14

.

8

&

15

.

5

\\

\newline

13

.

9

&

15

.

9

&

16

.

0

&

14

.

6

&

14

.

0

&

15

.

1

&

14

.

8

&

15

.

0

&

14

.

8

&

15

.

3

&

15

.

7

&

14

.

3

\newline

\end{tabular}

\newline

\begin{tabular}{llllllllllll}

\newline

13

.

9

&

15

.

6

&

15

.

4

&

14

.

6

&

15

.

2

&

14

.

8

&

13

.

7

&

16

.

3

&

15

.

1

&

14

.

5

&

13

.

6

&

15

.

1

\\

\newline

16

.

2

&

15

.

9

&

13

.

8

&

15

.

0

&

15

.

4

&

14

.

9

&

16

.

2

&

15

.

9

&

13

.

8

&

15

.

0

&

15

.

4

&

14

.

9

\\

\newline

14

.

2

&

16

.

5

&

13

.

4

&

16

.

5

&

14

.

8

&

15

.

1

&

14

.

9

&

13

.

7

&

16

.

2

&

15

.

8

&

15

.

4

&

14

.

7

\\

\newline

14

.

3

&

15

.

2

&

14

.

6

&

13

.

7

&

14

.

9

&

15

.

8

&

15

.

1

&

14

.

6

&

13

.

8

&

16

.

0

&

15

.

0

&

14

.

6

\newline

\end{tabular}

\newline

1

. Construct the grouped frequency distribution for the above given data using 'inclusive class intervals' and 'exclusive class intervals. Also report how many numbers of classes are required for the given data.

\newline

2

. Also construct Histogram and frequency curve.

Decide Number of Classes: Now, we need to decide on the number of classes for the frequency distribution. A common rule of thumb is to use the square root of the number of data points. $\newline$ Number of data points = $96$ $\newline$ Number of classes $\approx \sqrt{96}$ $\newline$ Number of classes $\approx 9.8$ $\newline$ Since we can't have a fraction of a class, we'll round up to $10$ classes.
Calculate Class Width: Next, we'll calculate the class width. Since we have $10$ classes and a range of $3.9$ , we'll divide the range by the number of classes. $\newline$ Class width $=$ Range $/$ Number of classes $\newline$ Class width $=$ $3.9 / 10$ $\newline$ Class width $=$ $0.39$ $\newline$ To make it easier, we'll round the class width up to $0.4$ .
Create Class Intervals: For inclusive class intervals, the lower class limit of the first class will be the smallest value, and each subsequent class will start with the next value after the upper limit of the previous class. $\newline$ First class interval: $13.2 - 13.6$ $\newline$ We'll continue creating intervals by adding the class width to the upper limit of the previous class.
Construct Histogram: For exclusive class intervals, the lower class limit of the first class will be the smallest value, but the upper limit will be one less than the next whole number. $\newline$ First class interval: $13.2 - 13.9$ $\newline$ We'll continue creating intervals by adding the class width to the upper limit of the previous class.
Construct Frequency Curve: Now, we'll construct the histogram. Each bar will represent a class interval, and the height of the bar will represent the frequency of data points within that interval. $\newline$ We'll count the number of data points in each interval and draw the bars accordingly.
Construct Frequency Curve: Now, we'll construct the histogram. Each bar will represent a class interval, and the height of the bar will represent the frequency of data points within that interval. We'll count the number of data points in each interval and draw the bars accordingly.Finally, we'll construct the frequency curve by plotting the midpoints of each class interval on the $x$ -axis and the corresponding frequencies on the $y$ -axis. Then, we'll connect the points with a smooth curve.

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