A large number of children were surveyed. The table below shows the distribution of the time they spent watching TV in the week before the survey. Intervals include the left endpoint but not the right. You can assume that the times are uniformly distributed within each interval. The table is used to draw a histogram of the distribution.
time (hours) percent
0-4 28
4-8 36
8-12 12
12-20 12
20-28 12
Find the height of the bar over interval 4-8, in percent per hour. Please do not enter units of measurement.
Estimate the 80th percentile of the distribution of times, in hours; give the best estimate you can, based on the information in the problem. Please do not enter units of measurement.
To find the height of the bar over the interval 4-8, we need to calculate the percentage per hour.
First, note that the interval 4-8 has a width of 4 hours (8 - 4).
Next, we calculate the total percentage of the interval by summing the percentages in the interval 4-8:
total percentage = 36%
Finally, we divide the total percentage by the width of the interval to get the height of the bar over the interval 4-8:
height of the bar = total percentage / width = 36% / 4 = 9% per hour.
Therefore, the height of the bar over the interval 4-8 is 9% per hour.
Moving on to estimating the 80th percentile of the distribution of times, we can see from the table that the percentages are divided into intervals and not uniformly distributed within each interval. However, we can still make an estimate based on the information given.
Since the intervals have equal widths and we know that the times are uniformly distributed within each interval, we can assume that the 80th percentile falls within the interval 8-12.
To estimate the exact time in hours, we can use linear interpolation within the interval from 8 to 12.
The 80th percentile corresponds to the cumulative percentage of time at or below the 80th percentile. In the given distribution, we can see that the cumulative percentage goes from 28% (0-4) to 64% (0-8) and then increases to 76% (0-12).
To find the best estimate of the 80th percentile within the 8-12 interval, we can set up a proportion:
(80 - 64) / (76 - 64) = (x - 8) / (12 - 8)
Solving for x, we find:
(x - 8) = (12 - 8) * (80 - 64) / (76 - 64)
x - 8 = 4 * 16 / 12
x - 8 = 64 / 12
x = 8 + 64 / 12
x = 8 + 5.33
So the best estimate for the 80th percentile of the distribution of times is approximately 13.33 hours.
Please note that this estimate is based solely on the information provided and makes assumptions about the distribution within each interval.