Question

The miles per gallon (mpg) for each of 20 medium-sized cars selected from a production line during the month of March follow. $$ \begin{array}{llll} 23.1 & 21.3 & 23.6 & 23.7 \\ 20.2 & 24.4 & 25.3 & 27.0 \\ 24.7 & 22.7 & 26.2 & 23.2 \\ 25.9 & 24.7 & 24.4 & 24.2 \\ 24.9 & 22.2 & 22.9 & 24.6 \end{array} $$ a. What are the maximum and minimum miles per gallon? What is the range? b. Construct a relative frequency histogram for these data. How would you describe the shape of the distribution? c. Find the mean and the standard deviation. d. Arrange the data from smallest to largest. Find the \(z\) -scores for the largest and smallest observations. Would you consider them to be outliers? Why or why not? e. What is the median? f. Find the lower and upper quartiles.

Short answer

Answer: The left-skewed distribution of the relative frequency histogram for the given mpg data indicates that there are more medium-sized cars with high mpg values than low mpg values, suggesting better fuel efficiency for a majority of these cars.

Step-by-step solution

Maximum, Minimum, and Range

First, list all the given miles per gallon values in ascending order: 20.2, 21.3, 22.2, 22.7, 22.9, 23.1, 23.2, 23.6, 23.7, 24.2, 24.4, 24.4, 24.6, 24.7, 24.7, 24.9, 25.3, 25.9, 26.2, 27.0 The minimum value is 20.2, and the maximum value is 27.0. The range is calculated as: Range = Maximum - Minimum = 27.0 - 20.2 = 6.8

Relative Frequency Histogram

To create a relative frequency histogram, we need to divide the data into intervals, count the frequency of each interval, and find the relative frequencies by dividing the interval frequencies by the total number of observations (20 cars). We can use 5 intervals as follows: 1. 20 - 21.9 2. 22 - 23.9 3. 24 - 25.9 4. 26 - 27 Calculate the frequency and relative frequency of each interval: 1. 20 - 21.9: Frequency = 2, Relative Frequency = 2/20 = 0.10 2. 22 - 23.9: Frequency = 7, Relative Frequency = 7/20 = 0.35 3. 24 - 25.9: Frequency = 10, Relative Frequency = 10/20 = 0.50 4. 26 - 27: Frequency = 1, Relative Frequency = 1/20 = 0.05 Based on the relative frequency histogram, the data appears to be slightly skewed to the left.

Mean and Standard Deviation

To find the mean, sum up all the mph values and divide by 20 (total number of cars): Mean = (sum of all mpg values) / 20 = 483.6 / 20 = 24.18 Now, calculate the standard deviation using the following formula: $$SD = \sqrt{\frac{\sum(x_i-\mu)^2}{n}}$$, where \(x_i\) are the mph values, \(\mu\) is the mean, and n is the number of cars. Standard deviation = 1.4838

Z-Scores and Outliers

Reorder the data from smallest to largest, as done in step 1. Now calculate the z-scores for the smallest and largest data points using formula: $$z = \frac{x - \mu}{SD}$$, where x is the data point, \(\mu\) is the mean, and SD is the standard deviation. For the smallest data point: z_min = (20.2 - 24.18) / 1.4838 = -2.679 For the largest data point: z_max = (27.0 - 24.18) / 1.4838 = 1.902 Both z-scores are not extremely large or small, so we would not consider them outliers.

Median

To find the median, we need to locate the middle value(s) in the ordered data. Since there are 20 cars (even number), the median is the average of the 10th and 11th ordered data points: Median = (24.2 + 24.4) / 2 = 24.3

Lower and Upper Quartiles

To find the lower quartile (Q1), locate the median of the lower half of the data, i.e., the data points from 1st to the 10th (inclusive). The lower quartile is the average of the 5th and 6th data points: Q1 = (22.9 + 23.1) / 2 = 23.0 To find the upper quartile (Q3), locate the median of the upper half of the data, i.e., the data points from the 11th to the 20th (inclusive). The upper quartile is the average of the 15th and 16th data points: Q3 = (24.7 + 24.9) / 2 = 24.8.

Key concepts

Histogram

A histogram is a graphical representation of data that uses bars to show the frequency of data within certain intervals. It's a great way to see the distribution and shape of the data.

To create a relative frequency histogram, you first divide your data into equal intervals or "bins." Count how many data points fall into each bin. For relative frequency, divide these counts by the total number of data points.

In the exercise, we have the following intervals:

• 20 - 21.9
• 22 - 23.9
• 24 - 25.9
• 26 - 27

For each interval, you calculate the frequency and then the relative frequency (e.g., 0.10, 0.35, etc.). This helps us understand how the data is spread across different intervals and whether it is skewed to one side.

The resulting histogram reveals that the data is slightly skewed to the left, indicating more data points at the higher end of the range.

Mean and Standard Deviation

The mean is the average of your data set and represents the central value. To calculate the mean, add all the values together and divide by the number of data points.

For our 20 car mpg values, the mean is calculated as:
\[ \text{Mean} = \frac{\sum \text{mpg values}}{20} = 24.18 \]

The standard deviation measures how spread out the values are from the mean. It's like a measure of the average distance each data point is from the mean.
The standard deviation formula is:
\[ SD = \sqrt{\frac{\sum(x_i - \mu)^2}{n}} \]
Where \(x_i\) are the data points and \(\mu\) is the mean.

In this exercise, the standard deviation was found to be approximately 1.4838, indicating that most values are close to the mean but have some variation.

Z-Scores

Z-scores help identify how many standard deviations a data point is from the mean. It's a way to determine if a particular data point is unusual or an outlier.

The formula for calculating a z-score is:
\[ z = \frac{x - \mu}{SD} \]
Where \(x\) is the data point, \(\mu\) is the mean, and \(SD\) is the standard deviation.

In this context, the smallest value (20.2) has a z-score of -2.679 and the largest value (27.0) has a z-score of 1.902.

A z-score typically indicates an outlier if it is less than -3 or greater than 3. Since neither of these z-scores meets that criterion, the values are not considered outliers.

Quartiles

Quartiles divide your data into four equal parts. They help to understand the spread and concentration of your data.

The lower quartile (Q1) is the median of the lower half of the data, and the upper quartile (Q3) is the median of the upper half.

To find Q1, look at the lower half of the ordered data. In our case, Q1 is the average of the 5th and 6th smallest data points:
\[ Q1 = \frac{22.9 + 23.1}{2} = 23.0 \]

For Q3, examine the upper half:
\[ Q3 = \frac{24.7 + 24.9}{2} = 24.8 \]

These calculations help in creating a box plot and identifying potential outliers, as well as understanding where most data points concentrate.