Approximate the mean of the grouped data

Approximate the mean of the grouped data. 11) Miles (per day) Frequency 1-2 29 3-4 12 5-6 18 7-8 2 9-10 16 A) 6 B) 15 C) 5 D 12) Weight (in pounds) Frequency 135-139 19 140-144 11 145-149 5 150-154 12 155-159 14 A) 144 B) 146 C) 148 D

The Correct Answer and Explanation is:

Here are the correct answers and a detailed explanation for each problem.

11) C) 5
12) B) 146

To approximate the mean of grouped data, a specific procedure is followed because the exact data values are unknown. We only know the intervals or bins they fall into. The method involves using the midpoint of each interval as a representative value for all the data points within that group.

The formula for the approximate mean is:
Mean ≈ Σ (f * x) / Σf
Where:

• f is the frequency of each group (the number of data points in that interval).
• x is the midpoint of each group’s interval.
• Σf is the sum of all frequencies, which represents the total number of data points.
• Σ(f * x) is the sum of the products of each midpoint and its corresponding frequency.

Explanation for Question 11

First, we calculate the midpoint for each “Miles (per day)” interval and multiply it by its frequency.

1. Find the midpoint (x) for each interval:
   • Interval 1-2: Midpoint = (1 + 2) / 2 = 1.5
   • Interval 3-4: Midpoint = (3 + 4) / 2 = 3.5
   • Interval 5-6: Midpoint = (5 + 6) / 2 = 5.5
   • Interval 7-8: Midpoint = (7 + 8) / 2 = 7.5
   • Interval 9-10: Midpoint = (9 + 10) / 2 = 9.5
2. Multiply each midpoint (x) by its frequency (f):
   • 1.5 * 29 = 43.5
   • 3.5 * 12 = 42
   • 5.5 * 18 = 99
   • 7.5 * 2 = 15
   • 9.5 * 16 = 152
3. Sum the results (Σ(f * x)) and the frequencies (Σf):
   • Sum of (f * x) = 43.5 + 42 + 99 + 15 + 152 = 351.5
   • Sum of frequencies (f) = 29 + 12 + 18 + 2 + 16 = 77
4. Calculate the approximate mean:
   • Mean ≈ 351.5 / 77 ≈ 4.56
   • The calculated mean of 4.56 is closest to the answer choice C) 5.

Explanation for Question 12

We apply the same method to the “Weight (in pounds)” data.

1. Find the midpoint (x) for each interval:
   • Interval 135-139: Midpoint = (135 + 139) / 2 = 137
   • Interval 140-144: Midpoint = (140 + 144) / 2 = 142
   • Interval 145-149: Midpoint = (145 + 149) / 2 = 147
   • Interval 150-154: Midpoint = (150 + 154) / 2 = 152
   • Interval 155-159: Midpoint = (155 + 159) / 2 = 157
2. Multiply each midpoint (x) by its frequency (f):
   • 137 * 19 = 2603
   • 142 * 11 = 1562
   • 147 * 5 = 735
   • 152 * 12 = 1824
   • 157 * 14 = 2198
3. Sum the results (Σ(f * x)) and the frequencies (Σf):
   • Sum of (f * x) = 2603 + 1562 + 735 + 1824 + 2198 = 8922
   • Sum of frequencies (f) = 19 + 11 + 5 + 12 + 14 = 61
4. Calculate the approximate mean:
   • Mean ≈ 8922 / 61 ≈ 146.26
   • The calculated mean of 146.26 is closest to the answer choice B) 146.