Certainly! Let's go through the steps to address each part of the question based on the given data and results.
### 7.1 Calculate the mean and standard deviation of the given data.
To calculate the mean and standard deviation, we use the formulas:
- Mean [tex]\((\mu)\)[/tex] = [tex]\(\frac{\sum X}{N}\)[/tex], where [tex]\(X\)[/tex] is each data value and [tex]\(N\)[/tex] is the number of data values.
- Standard Deviation [tex]\((\sigma)\)[/tex] = [tex]\(\sqrt{\frac{\sum (X - \mu)^2}{N}}\)[/tex].
Based on the given data, the calculations yield:
- Mean: 14.407333333333334
- Standard Deviation: 2.0528560321875693
### 7.2 Complete the cumulative frequency column in the table provided.
To complete the cumulative frequency, we count the number of observations up to and including each data value, gradually increasing the cumulative total.
The cumulative frequencies for the given data set are as follows (the format is [tex]\( (data\ point,\ cumulative\ frequency) \)[/tex]):
- (11, 1)
- (11.9, 2)
- (12, 3)
- (12.52, 4)
- (13.125, 5)
- (13.125, 6)
- (13.15, 7)
- (13.2, 8)
- (13.25, 9)
- (13.25, 10)
- (13.25, 11)
- (13.25, 12)
- (13.4, 13)
- (13.5, 14)
- (13.7, 15)
- (13.7, 16)
- (14, 17)
- (14.2, 18)
- (14.2, 19)
- (14.5, 20)
- (14.5, 21)
- (15, 22)
- (15.5, 23)
- (15.6, 24)
- (16.1, 25)
- (17.3, 26)
- (17.5, 27)
- (17.8, 28)
- (19, 29)
- (19.7, 30)
### 7.3 Draw an Ogive (cumulative frequency curve) of the given data on the grid provided.
To plot an Ogive, simply plot the cumulative frequencies on the y-axis against the corresponding data values on the x-axis, and connect the points with a smooth curve.
### 7.4 Draw the box and whisker diagram for the given data.
A box and whisker plot includes the following elements:
- Minimum value (min): 11.0
- First quartile (Q1): 13.2125
- Median (Q2): 13.7
- Third quartile (Q3): 15.375
- Maximum value (max): 19.7
Using these values, plot the following:
1. Draw a number line that covers the range of the data.
2. Draw a box from Q1 (13.2125) to Q3 (15.375).
3. Draw a line inside the box at the median (13.7).
4. Draw whiskers (lines) from Q1 to the minimum value (11.0) and from Q3 to the maximum value (19.7).
### 7.5 Comment on the skewness of the data.
The skewness of a dataset can be determined by looking at the relative positions of the quartiles and the whiskers. If the upper whisker is noticeably longer than the lower whisker, the data is right-skewed.
In this case, since the maximum value (19.7) is much further from Q3 (15.375) compared to how close the Q1 (13.2125) is to the minimum value (11.0), we can conclude that the data is right-skewed.
