A researcher conducts a mileage economy test involving 60 cars. The frequency distribution describing average miles per gallon (mpg) appears in the following table.
Average mpg Frequency
15 < x ≤ 20 3
20 < x ≤ 25 14
25 < x ≤ 30 13
30 < x ≤ 35 4
35 < x ≤ 40 3
40 < x ≤ 45 23
a-1. Construct the relative frequency distribution and the cumulative relative frequency distribution. (Round your answers to 4 decimal places.)
a-2. What proportion of the cars got more than 25 mpg but no more than 30 mpg? (Round your answer to 4 decimal places.)
a-3. What proportion of the cars got 30 mpg or less? (Round your answer to 4 decimal places.)
a-4. What proportion of the cars got more than 30 mpg? (Round your answer to 4 decimal places.)
b. Which statement is correct regarding the shape of the distribution using a histogram?
multiple choice
The distribution is symmetric.
The distribution is not symmetric.
a-1. To construct the relative frequency distribution, divide the frequency of each class by the total number of cars (60). To construct the cumulative relative frequency distribution, sum up the relative frequencies.
Average mpg Frequency Relative Frequency Cumulative Relative Frequency
15 < x ≤ 20 3 0.05 0.05
20 < x ≤ 25 14 0.2333 0.2833
25 < x ≤ 30 13 0.2167 0.5
30 < x ≤ 35 4 0.0667 0.5667
35 < x ≤ 40 3 0.05 0.6167
40 < x ≤ 45 23 0.3833 1
a-2. The proportion of cars that got more than 25 mpg but no more than 30 mpg can be found by summing up the relative frequencies of those categories (25 < x ≤ 30 and 30 < x ≤ 35): 0.2167 + 0.0667 = 0.2834.
a-3. The proportion of cars that got 30 mpg or less is the cumulative relative frequency for the category 30 < x ≤ 35, which is 0.5667.
a-4. The proportion of cars that got more than 30 mpg can be found by subtracting the cumulative relative frequency of 30 < x ≤ 35 from 1 (total proportion): 1 - 0.5667 = 0.4333.
b. According to the given data, we cannot determine whether the distribution is symmetric or not.
a-1. To construct the relative frequency distribution, divide the frequency of each class by the total number of cars (60). Then, calculate the cumulative relative frequency by adding up the relative frequencies.
Average mpg | Frequency | Relative Frequency | Cumulative Relative Frequency
------------------|-------------|----------------------------|------------------------------------
15 < x ≤ 20 | 3 | 3/60 = 0.0500 | 0.0500
20 < x ≤ 25 | 14 | 14/60 = 0.2333 | 0.2833
25 < x ≤ 30 | 13 | 13/60 = 0.2167 | 0.5000
30 < x ≤ 35 | 4 | 4/60 = 0.0667 | 0.5667
35 < x ≤ 40 | 3 | 3/60 = 0.0500 | 0.6167
40 < x ≤ 45 | 23 | 23/60 = 0.3833 | 1.0000
a-2. The proportion of the cars that got more than 25 mpg but no more than 30 mpg can be calculated by summing up the relative frequencies of those classes.
Proportion = Relative Frequency (25 < x ≤ 30) = 0.2167
a-3. The proportion of cars that got 30 mpg or less can be calculated by summing up the relative frequencies of the classes up to and including the 30 < x ≤ 35 class.
Proportion = Cumulative Relative Frequency (30 < x ≤ 35) = 0.5667
a-4. The proportion of cars that got more than 30 mpg can be calculated by subtracting the proportion in a-3 from 1.
Proportion = 1 - Proportion (30 mpg or less) = 1 - 0.5667 = 0.4333
b. The statement "The distribution is not symmetric" is correct regarding the shape of the distribution using a histogram.