It seems these days that students have to work more and more each week to be able to afford the costs of college. For a recent labor survey conducted by the government, there were
79
respondents who were working while attending college. The frequency distribution below summarizes the number of hours worked per week as reported by the respondents.
Number of hours
worked per week
(hours per week) Frequency
26
to
30 9
31
to
35 26
36
to
40 20
41
to
45 12
46
to
50 7
51
to
55 5
Based on the frequency distribution, using the midpoint of each data class, estimate the mean number of hours worked per week by the respondents. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.
To estimate the mean number of hours worked per week, we need to calculate the midpoint of each data class and then find the weighted average.
The midpoint of each interval is calculated by adding the lower and upper boundaries of each interval and dividing by 2.
Midpoint for 26 to 30 hours per week = (26+30)/2 = 28
Midpoint for 31 to 35 hours per week = (31+35)/2 = 33
Midpoint for 36 to 40 hours per week = (36+40)/2 = 38
Midpoint for 41 to 45 hours per week = (41+45)/2 = 43
Midpoint for 46 to 50 hours per week = (46+50)/2 = 48
Midpoint for 51 to 55 hours per week = (51+55)/2 = 53
Next, we multiply each midpoint by its respective frequency and sum the results:
(28*9) + (33*26) + (38*20) + (43*12) + (48*7) + (53*5) = 1186
Finally, we divide the sum by the total number of respondents (79):
Mean number of hours worked per week = 1186/79 ≈ 15
Therefore, the estimated mean number of hours worked per week by the respondents is approximately 15.
To estimate the mean number of hours worked per week by the respondents, we need to calculate the midpoint of each data class and then find the weighted average of these midpoints.
To find the midpoint of each data class, we can use the formula:
Midpoint = (lower class limit + upper class limit) / 2
Using this formula, we can calculate the midpoints for each data class:
Midpoint for 26 to 30 hours per week = (26 + 30) / 2 = 28
Midpoint for 31 to 35 hours per week = (31 + 35) / 2 = 33
Midpoint for 36 to 40 hours per week = (36 + 40) / 2 = 38
Midpoint for 41 to 45 hours per week = (41 + 45) / 2 = 43
Midpoint for 46 to 50 hours per week = (46 + 50) / 2 = 48
Midpoint for 51 to 55 hours per week = (51 + 55) / 2 = 53
Now, let's calculate the weighted average of these midpoints using the formula:
Mean = (Sum of (midpoint * frequency)) / (Sum of frequencies)
Mean = (28 * 9 + 33 * 26 + 38 * 20 + 43 * 12 + 48 * 7 + 53 * 5) / (9 + 26 + 20 + 12 + 7 + 5)
Mean = (252 + 858 + 760 + 516 + 336 + 265) / 79
Mean = 2987 / 79
Mean ≈ 37.8
Therefore, the estimated mean number of hours worked per week by the respondents is approximately 37.8 hours.