Question

The miles per gallon (mpg) for each of 20 medium-sized cars selected from a production line during the month of March follow. \(\begin{array}{llll}23.1 & 21.3 & 23.6 & 23.7\end{array}\) \(\begin{array}{llll}20.2 & 24.4 & 25.3 & 27.0 \\ 24.7 & 22.7 & 26.2 & 23.2\end{array}\) 25.9 \(\begin{array}{llll}24.9 & 22.2 & 22.9 & 24.6\end{array}\) 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 range for the mpg data set is 6.8, the standard deviation is approximately 2.8154, and the z-score of the smallest observation is approximately -1.3756.

Step-by-step solution

a. Maximum, Minimum, and Range

First, we need to find the maximum and minimum values of miles per gallon (mpg). After that, we'll calculate the range by subtracting the minimum value from the maximum value. The maximum mpg = 27.0 The minimum mpg = 20.2 Range = Maximum mpg - Minimum mpg = 27.0 - 20.2 = 6.8

b. Relative frequency histogram and distribution shape

To construct a relative frequency histogram, we need to organize the data into intervals (or bins) and calculate the relative frequency for each interval. For this exercise, we will use intervals of size 2, starting from 20. Then, we can construct a histogram by plotting the intervals (or bins) along the x-axis and the corresponding relative frequencies along the y-axis. Here is the relative frequency table for the given data: Interval | Frequency | Relative Frequency 20-22 | 3 | 3 / 20 = 0.15 22-24 | 4 | 4 / 20 = 0.20 24-26 | 9 | 9 / 20 = 0.45 26-28 | 4 | 4 / 20 = 0.20 The histogram will have a peak around 24-26, and the shape of the distribution can be described as roughly unimodal and slightly right-skewed.

c. Mean and Standard Deviation

To calculate the mean, we add up all the mpg values and divide the sum by the total number of cars (20): Mean = (23.1 + 21.3 + 23.6 + ... + 22.9 + 24.6) / 20 = 481.5 / 20 = 24.075 To calculate the standard deviation, first, find the difference between each value and the mean, square those differences, and find the average of those squared differences. Finally, take the square root of that average: Variance = [(23.1-24.075)^2 + (21.3-24.075)^2 + ... + (22.9-24.075)^2 + (24.6-24.075)^2] / 20 Variance ≈ 7.9224 Standard Deviation = √Variance ≈ √7.9224 ≈ 2.8154

d. Z-scores for the largest and smallest observations and outliers

First, arrange the data from smallest to largest: 20.2, 21.3, 22.2, 22.7, 22.9, 23.1, 23.2, 23.6, 23.7, 24.4, 24.6, 24.7, 24.9, 25.3, 25.9, 26.2, 27.0 Now, we'll find the z-scores for the largest (27.0) and smallest (20.2) observations: z-score = (Observation - Mean) / Standard Deviation z-score (smallest) ≈ (20.2 - 24.075) / 2.8154 ≈ -1.3756 z-score (largest) ≈ (27.0 - 24.075) / 2.8154 ≈ 1.0399 Since both z-scores, -1.3756 and 1.0399, are not extreme values (typically, values that are more than 2 or 3 standard deviations away from the mean are considered extreme), we would not consider them to be outliers.

e. Median

To find the median, we need to locate the middle value when the data is arranged in ascending order. In this case, we have 20 values, so the median will be the average of the 10th and 11th values: Median = (24.4 + 24.6) / 2 = 24.5

f. Lower and Upper Quartiles

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 of the data. To find Q1, we'll find the median of the first 10 values: Q1 = (22.7 + 22.9) / 2 = 22.8 To find Q3, we'll find the median of the last 10 values: Q3 = (25.3 + 25.9) / 2 = 25.6

Key concepts

Mean

The mean, often referred to as the average, is a measure of central tendency that gives us an idea about the central point of a data set. To calculate the mean for a data set, add up all the data points and divide the total by the number of data points. In our example of the miles per gallon (mpg) for cars, we sum the mpg values and divide by 20, as there are 20 cars in the sample. The mean gives us a quick estimate of the overall fuel efficiency of this group of cars.
The formula to find the mean is:\[\text{Mean} = \frac{\sum X}{N}\]where \(\sum X\) is the sum of all values, and \(N\) is the number of values. Understanding the mean helps in comparing the efficiency of different models or batches by providing one central value for comparison.

Standard Deviation

Standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of values. A low standard deviation means the data points are close to the mean, whereas a high standard deviation indicates the data points are spread out over a broader range.
To calculate standard deviation:

• Find the mean of the data set.
• Subtract the mean from each data value and square the result.
• Calculate the mean of these squared differences.
• Take the square root of this average.

The formula for standard deviation is:\[\sigma = \sqrt{\frac{\sum (X - \text{Mean})^2}{N}}\]In our mpg example, a standard deviation of approximately 2.8154 indicates how much each car's mpg might differ from the average mpg of 24.075. This measure supports decision-making processes, such as identifying how typical or atypical a car's performance is relative to others in the group.

Median

The median is another central tendency measure that represents the middle point of a data set, essentially splitting the higher half from the lower half. To find the median, first arrange all data points in ascending order. If there is an odd number of observations, the median is the middle number. With an even number, as in our case with 20 mpg values, we take the average of the two central numbers.
For our arranged mpg data, the median was calculated as the average of the 10th and 11th values, which were 24.4 and 24.6, providing us a median of 24.5. The median can be a more robust indicator than the mean in cases where the data set includes outliers or is skewed, as the median is not affected by extremely high or low values.

Quartiles

Quartiles divide a ranked data set into four equal parts, providing insights into the distribution of data. The lower quartile (Q1) marks the 25th percentile, the median (Q2) marks the 50th, and the upper quartile (Q3) marks the 75th percentile. They help reveal the spread and identify any skewness in the data.
In our mpg data, after arranging the values, we find the lower quartile by taking the median of the first half (first 10 values), resulting in Q1 = 22.8. Similarly, the upper quartile is the median of the second half (last 10 values), resulting in Q3 = 25.6.

• Q1 shows the point below which 25% of data lies.
• Q3 shows the point below which 75% of data lies.

The interquartile range (IQR), calculated as Q3 - Q1, helps measure the statistical dispersion between these quartiles.