Consider the following frequency distribution:
Interval Frequency
10 < x ≤ 20 30
20 < x ≤ 30 37
30 < x ≤ 40 48
40 < x ≤ 50 25
a-1. Construct the relative frequency distribution. (Round your answers to 3 decimal places.)
a-2. What proportion of the observations are in the interval 20 < x ≤ 30?
b-1. Construct the cumulative frequency distribution and the cumulative relative frequency distribution. (Round "Cumulative Relative Frequency" to 3 decimal places.)
b-2. How many of the observations are less than 40?
c. What proportion of the observations are less than 40? (Round your answer to 1 decimal place.)
a-1. To construct the relative frequency distribution, divide each frequency by the sum of all the frequencies.
Interval Frequency Relative Frequency
10 < x ≤ 20 30 0.222
20 < x ≤ 30 37 0.274
30 < x ≤ 40 48 0.356
40 < x ≤ 50 25 0.185
a-2. The proportion of the observations in the interval 20 < x ≤ 30 is equal to its relative frequency, which is 0.274.
b-1. To construct the cumulative frequency distribution, add up the frequencies as you go down the list. The cumulative relative frequency is found by dividing the cumulative frequency by the sum of all frequencies.
Interval Frequency Cumulative Frequency Cumulative Relative Frequency
10 < x ≤ 20 30 30 0.222
20 < x ≤ 30 37 67 0.496
30 < x ≤ 40 48 115 0.852
40 < x ≤ 50 25 140 1.000
b-2. The cumulative frequency of the observations less than 40 is equal to the cumulative frequency of the interval 30 < x ≤ 40, which is 115.
c. The proportion of the observations less than 40 is equal to the cumulative relative frequency of the interval 30 < x ≤ 40, which is 0.852.
a-1. To construct the relative frequency distribution, divide each frequency by the total number of observations.
First, calculate the total number of observations:
Total number of observations = 30 + 37 + 48 + 25 = 140
Now, divide each frequency by 140 to get the relative frequency:
Interval Frequency Relative Frequency
10 < x ≤ 20 30 30/140 = 0.214
20 < x ≤ 30 37 37/140 ≈ 0.264
30 < x ≤ 40 48 48/140 = 0.343
40 < x ≤ 50 25 25/140 ≈ 0.179
a-2. The proportion of the observations in the interval 20 < x ≤ 30 is equal to the relative frequency for that interval. From the relative frequency distribution, we can see that the relative frequency for this interval is approximately 0.264.
b-1. To construct the cumulative frequency distribution, add up the frequencies for each interval as we go:
Interval Frequency Cumulative Frequency Cumulative Relative Frequency
10 < x ≤ 20 30 30 30/140 = 0.214
20 < x ≤ 30 37 30 + 37 = 67 67/140 ≈ 0.479
30 < x ≤ 40 48 67 + 48 = 115 115/140 ≈ 0.821
40 < x ≤ 50 25 115 + 25 = 140 140/140 = 1.000
b-2. From the cumulative frequency distribution, we can see that the cumulative frequency for the interval less than 40 is 115.
c. To find the proportion of the observations less than 40, we can divide the cumulative frequency for that interval by the total number of observations:
Proportion of observations less than 40 = Cumulative frequency for interval < 40 / Total number of observations
= 115 / 140 ≈ 0.821 (rounded to one decimal place)