A researcher conducts a mileage economy test involving 72 cars. The frequency distribution describing average miles per gallon (mpg) appears in the following table.
Average mpg Frequency
15 < x ≤ 20 25
20 < x ≤ 25 5
25 < x ≤ 30 8
30 < x ≤ 35 11
35 < x ≤ 40 13
40 < x ≤ 45 10
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 each frequency by the total number of cars (72) and round to four decimal places:
Average mpg Frequency Relative Frequency
15 < x ≤ 20 25 25/72 = 0.3472
20 < x ≤ 25 5 5/72 = 0.0694
25 < x ≤ 30 8 8/72 = 0.1111
30 < x ≤ 35 11 11/72 = 0.1528
35 < x ≤ 40 13 13/72 = 0.1806
40 < x ≤ 45 10 10/72 = 0.1389
To construct the cumulative relative frequency distribution, add up the relative frequencies starting from the first class:
Average mpg Frequency Relative Frequency Cumulative Relative Frequency
15 < x ≤ 20 25 0.3472 0.3472
20 < x ≤ 25 5 0.0694 0.4166
25 < x ≤ 30 8 0.1111 0.5277
30 < x ≤ 35 11 0.1528 0.6805
35 < x ≤ 40 13 0.1806 0.8611
40 < x ≤ 45 10 0.1389 1.0000
a-2. The proportion of cars that got more than 25 mpg but no more than 30 mpg can be found by subtracting the cumulative relative frequency up to 25 mpg from the cumulative relative frequency up to 30 mpg:
Proportion = Cumulative Relative Frequency (up to 30 mpg) - Cumulative Relative Frequency (up to 25 mpg)
= 0.6805 - 0.5277
= 0.1528
Therefore, the proportion is 0.1528.
a-3. The proportion of cars that got 30 mpg or less is simply the cumulative relative frequency up to 30 mpg:
Proportion = Cumulative Relative Frequency (up to 30 mpg)
= 0.6805
Therefore, the proportion is 0.6805.
a-4. The proportion of cars that got more than 30 mpg can be found by subtracting the cumulative relative frequency up to 30 mpg from 1:
Proportion = 1 - Cumulative Relative Frequency (up to 30 mpg)
= 1 - 0.6805
= 0.3195
Therefore, the proportion is 0.3195.
b. The statement "The distribution is not symmetric" is correct regarding the shape of the distribution using a histogram as the distribution is likely to be right-skewed.
a-1. To construct the relative frequency distribution, divide each frequency by the total number of cars (72).
Average mpg Frequency Relative Frequency
15 < x ≤ 20 25 25/72 ≈ 0.3472
20 < x ≤ 25 5 5/72 ≈ 0.0694
25 < x ≤ 30 8 8/72 ≈ 0.1111
30 < x ≤ 35 11 11/72 ≈ 0.1528
35 < x ≤ 40 13 13/72 ≈ 0.1806
40 < x ≤ 45 10 10/72 ≈ 0.1389
To construct the cumulative relative frequency distribution, add up the relative frequencies from the start up to each category.
Average mpg Cumulative Relative Frequency
15 < x ≤ 20 0.3472
20 < x ≤ 25 0.4166 (0.3472 + 0.0694)
25 < x ≤ 30 0.5277 (0.4166 + 0.1111)
30 < x ≤ 35 0.6805 (0.5277 + 0.1528)
35 < x ≤ 40 0.8611 (0.6805 + 0.1806)
40 < x ≤ 45 1.0000 (0.8611 + 0.1389)
a-2. The proportion of cars that got more than 25 mpg but no more than 30 mpg is equal to the relative frequency of the category 25 < x ≤ 30. Therefore, the answer is 0.1111.
a-3. The proportion of cars that got 30 mpg or less can be found by adding up the relative frequencies of the categories 15 < x ≤ 20, 20 < x ≤ 25, and 25 < x ≤ 30. Therefore, the answer is 0.3472 + 0.0694 + 0.1111 = 0.5277.
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. Therefore, the answer is 1 - 0.6805 = 0.3195.
b. The statement "The distribution is not symmetric" is correct regarding the shape of the distribution using a histogram.