Question

problem 6: completing and interpreting histograms
a survey tracked daily screen time (in hours) for high school students. the first 20 responses are shown in the histogram. the remaining 8 responses were: 6, 6, 7, 7, 8, 8, 9, 10
current histogram (first 20 responses):
additional 8 responses to add: 6, 6, 7, 7, 8, 8, 9, 10 hours
a. complete the histogram by adding the remaining 8 data points.
b. compare the mean and median for this complete distribution. which is greater and why?
key skills: create histograms and use distribution shape to compare measures of center

Explanation:

Step1: Count new data in intervals

For the 8 new data - points (6, 6, 7, 7, 8, 8, 9, 10), in the interval 6 - 7, there are 4 data - points, in the interval 8 - 9 there are 2 data - points, and in the interval 9 - 10 there is 1 data - point, and in the interval 10 - 11 there is 1 data - point. Add these frequencies to the existing histogram.

Step2: Find total number of data points

The total number of data points is \(20 + 8=28\).

Step3: Arrange all data points

Arrange all 28 data points in ascending order. We don't know the exact 20 data points from the histogram, but we combine them with the 8 given data points: 6, 6, 7, 7, 8, 8, 9, 10 and the 20 from the histogram.

Step4: Calculate the median

Since \(n = 28\) (an even - numbered data set), the median is the average of the \(\frac{n}{2}=14\)th and \((\frac{n}{2}+1) = 15\)th ordered data values.

Step5: Calculate the mean

Let the sum of the first 20 data points be \(S_1\) and the sum of the 8 new data points \(S_2=6 + 6+7 + 7+8 + 8+9 + 10=51\). The mean \(\bar{x}=\frac{S_1 + S_2}{28}\). In a right - skewed distribution (which is likely here as larger values like 9 and 10 can pull the mean), the mean is greater than the median because the mean is affected by extreme values more than the median.

Answer:

a. Add 4 to the frequency of the 6 - 7 interval, 2 to the 8 - 9 interval, 1 to the 9 - 10 interval and 1 to the 10 - 11 interval in the existing histogram.
b. The mean is greater than the median because the distribution is likely right - skewed and the mean is more influenced by the larger values (such as 9 and 10) than the median.