Consider the following relative frequency distribution:
Interval Relative Frequency
−20 < x ≤ −10 0.26
−10 < x ≤ 0 0.22
0 < x ≤ 10 0.04
10 < x ≤ 20 0.26
20 < x ≤ 30 0.22
a-1. Suppose this relative frequency distribution is based on a sample of 48 observations. Construct the frequency distribution.
a-2. How many of the observations are more 10 but no more than 20?
b-1. Construct the cumulative frequency distribution.
b-2. How many of the observations are −10 or less?
a-1. To construct the frequency distribution, we need to multiply each relative frequency by the total number of observations.
Interval Relative Frequency Frequency
−20 < x ≤ −10 0.26 0.26 * 48 = 12.48
−10 < x ≤ 0 0.22 0.22 * 48 = 10.56
0 < x ≤ 10 0.04 0.04 * 48 = 1.92
10 < x ≤ 20 0.26 0.26 * 48 = 12.48
20 < x ≤ 30 0.22 0.22 * 48 = 10.56
Rounding the frequencies to whole numbers:
Interval Frequency
−20 < x ≤ −10 12
−10 < x ≤ 0 11
0 < x ≤ 10 2
10 < x ≤ 20 12
20 < x ≤ 30 11
a-2. The number of observations that are more than 10 but no more than 20 is equal to the frequency of the interval 10 < x ≤ 20:
Frequency = 12
b-1. To construct the cumulative frequency distribution, we add up the frequencies starting from the lowest interval to the highest interval.
Interval Frequency Cumulative Frequency
−20 < x ≤ −10 12 12
−10 < x ≤ 0 11 23
0 < x ≤ 10 2 25
10 < x ≤ 20 12 37
20 < x ≤ 30 11 48
b-2. The number of observations that are -10 or less is equal to the cumulative frequency of the interval -20 < x ≤ -10:
Cumulative Frequency = 12
a-1. To construct the frequency distribution, we need to calculate the frequency of each interval by multiplying the relative frequency by the total number of observations (48).
Interval Relative Frequency Frequency
-20 < x ≤ -10 0.26 0.26 * 48 = 12.48 ≈ 12
-10 < x ≤ 0 0.22 0.22 * 48 = 10.56 ≈ 11
0 < x ≤ 10 0.04 0.04 * 48 = 1.92 ≈ 2
10 < x ≤ 20 0.26 0.26 * 48 = 12.48 ≈ 12
20 < x ≤ 30 0.22 0.22 * 48 = 10.56 ≈ 11
The frequency distribution is as follows:
Interval Frequency
-20 < x ≤ -10 12
-10 < x ≤ 0 11
0 < x ≤ 10 2
10 < x ≤ 20 12
20 < x ≤ 30 11
a-2. To find the number of observations that are more than 10 but no more than 20, we sum up the frequencies of the corresponding intervals:
Number of observations = Frequency of (10 < x ≤ 20) = 12
b-1. To construct the cumulative frequency distribution, we sum up the frequencies as we move through the intervals.
Interval Frequency Cumulative Frequency
-20 < x ≤ -10 12 12
-10 < x ≤ 0 11 23
0 < x ≤ 10 2 25
10 < x ≤ 20 12 37
20 < x ≤ 30 11 48
b-2. To find the number of observations that are -10 or less, we look at the cumulative frequency of the interval (-20 < x ≤ -10):
Number of observations = Cumulative frequency of (-20 < x ≤ -10) = 12