Questions: The following is a set of data from a sample of n=5. 4, 3, 10, 7, 8 a. Compute the mean, median, and mode. b. Compute the range, variance, standard deviation, and coefficient of variation. c. Compute the Z scores. Are there any outliers? d. Describe the shape of the data set. A. The median is 7. (Type an integer or a decimal. Do not round. Use a comma to separate answers as needed) B. There is no solution. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mode is . (Type an integer or a decimal. Do not round. Use a comma to separate answers as needed) B. There is no solution. b. The range is 7. (Type an integer or a decimal. Do not round.) The variance is . (Type an integer or decimal rounded to two decimal places as needed.)

The following is a set of data from a sample of n=5.
4, 3, 10, 7, 8
a. Compute the mean, median, and mode.
b. Compute the range, variance, standard deviation, and coefficient of variation.
c. Compute the Z scores. Are there any outliers?
d. Describe the shape of the data set.
A. The median is 7.
(Type an integer or a decimal. Do not round. Use a comma to separate answers as needed)
B. There is no solution.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The mode is .
(Type an integer or a decimal. Do not round. Use a comma to separate answers as needed)
B. There is no solution.
b. The range is 7. (Type an integer or a decimal. Do not round.)

The variance is .
(Type an integer or decimal rounded to two decimal places as needed.)

Transcript text: The following is a set of data from a sample of $n=5$. \[ \begin{array}{lllll} 4 & 3 & 10 & 7 & 8 \end{array} \] a. Compute the mean, median, and mode. b. Compute the range, variance, standard deviation, and coefficient of variation. c. Compute the Z scores. Are there any outliers? d. Describe the shape of the data set. A. The median is 7 . (Type an integer or a decimal. Do not round. Use a comma to separate answers as needed) B. There is no solution. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mode is $\square$ (Type an integer or a decimal. Do not round. Use a comma to separate answers as needed) B. There is no solution. b. The range is 7 . (Type an integer or a decimal. Do not round.) The variance is $\square$. (Type an integer or decimal rounded to two decimal places as needed.)

Solution

Solution Steps

To solve the given problem, we will follow these steps:

a. Mean, Median, and Mode:

Calculate the mean by summing all the data points and dividing by the number of data points.
Find the median by sorting the data and selecting the middle value.
Determine the mode by identifying the most frequently occurring value(s).

b. Range, Variance, Standard Deviation, and Coefficient of Variation:

Compute the range by subtracting the smallest value from the largest value.
Calculate the variance by finding the average of the squared differences from the mean.
Determine the standard deviation as the square root of the variance.
Calculate the coefficient of variation by dividing the standard deviation by the mean and multiplying by 100.

c. Z Scores and Outliers:

Compute the Z score for each data point by subtracting the mean and dividing by the standard deviation.
Identify outliers as data points with Z scores greater than 3 or less than -3.

To solve the given problem, we will follow the steps outlined in the question. Let's begin with the calculations.

Step 1: Compute the Mean, Median, and Mode

Mean: The mean is the average of the data set. It is calculated as follows: \[ \text{Mean} = \frac{4 + 3 + 10 + 7 + 8}{5} = \frac{32}{5} = 6.4 \]
Median: The median is the middle value of the ordered data set. First, we order the data: \[ 3, 4, 7, 8, 10 \] The median is the third value, which is 7.
Mode: The mode is the value that appears most frequently. In this data set, each number appears only once, so there is no mode.

Step 2: Compute the Range, Variance, Standard Deviation, and Coefficient of Variation

Range: The range is the difference between the maximum and minimum values. \[ \text{Range} = 10 - 3 = 7 \]
Variance: The variance is the average of the squared differences from the mean. \[ \text{Variance} = \frac{(4-6.4)^2 + (3-6.4)^2 + (10-6.4)^2 + (7-6.4)^2 + (8-6.4)^2}{5} \] \[ = \frac{(2.4)^2 + (3.4)^2 + (3.6)^2 + (0.6)^2 + (1.6)^2}{5} \] \[ = \frac{5.76 + 11.56 + 12.96 + 0.36 + 2.56}{5} = \frac{33.2}{5} = 6.64 \]
Standard Deviation: The standard deviation is the square root of the variance. \[ \text{Standard Deviation} = \sqrt{6.64} \approx 2.5777 \]
Coefficient of Variation: The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage. \[ \text{Coefficient of Variation} = \left(\frac{2.5777}{6.4}\right) \times 100 \approx 40.27\% \]

Step 3: Compute the Z Scores and Identify Outliers

Z Scores: The Z score for each data point is calculated as follows: \[ Z = \frac{X - \text{Mean}}{\text{Standard Deviation}} \]
- For 4: $ Z = \frac{4 - 6.4}{2.5777} \approx -0.9309 $
- For 3: $ Z = \frac{3 - 6.4}{2.5777} \approx -1.3190 $
- For 10: $ Z = \frac{10 - 6.4}{2.5777} \approx 1.3964 $
- For 7: $ Z = \frac{7 - 6.4}{2.5777} \approx 0.2327 $
- For 8: $ Z = \frac{8 - 6.4}{2.5777} \approx 0.6209 $
Outliers: Typically, a Z score greater than 3 or less than -3 indicates an outlier. None of the Z scores are beyond this range, so there are no outliers.