Question

a random sample of 120 students was selected from those students who completed a survey in a general introductory statistics course. the survey asked the number of music cds owned by each of these students. the histogram of the number of music cds owned by the students is shown below. histogram and stem and leaf plot (labeled stem and leaf with stem, leaf, count columns; note 0|1 represents 10) shown which of the following statements does not describe the distribution of number of music cds owned?

• the median number of music cds owned is less than 50.
• the maximum number of cds owned is between 600 and 650.
• the mean number of music cds owned is greater than the median number of music cds owned.
• the distribution of music cds owned is skewed to the right.
• there are 98 students who own less than 150 music cds.

Explanation:

Step1: Analyze the stem - and - leaf plot and histogram

The stem - and - leaf plot and histogram show the distribution of the number of music CDs owned by 120 students. The distribution is right - skewed (tail on the right side with larger values). For a right - skewed distribution, the mean is greater than the median.

Step2: Check the median

To find the median, we need to find the middle value(s) of the 120 data points. The median will be the average of the 60th and 61st values. By looking at the counts in the stem - and - leaf plot:
- The first few rows: Count for stem 6:1, stem 5:1, stem 4:0, stem 3:2, stem 2:4, stem 1 (first row):10, stem 1 (second row):17, stem 0 (first row):29, stem 0 (second row):52.
- Let's sum the counts: 1 + 1+0 + 2+4 + 10+17+29+52=116. Wait, maybe a better way: The total number of students is 120. The median is the average of the 60th and 61st terms. Looking at the stem - and - leaf plot, the lower stems (smaller number of CDs) have a large number of counts. The cumulative count up to stem 0 (second row) is 52, then stem 0 (first row) is 29, so cumulative up to stem 0 (first row) is 52 + 29=81. Then stem 1 (second row) is 17, cumulative 81+17 = 98. Stem 1 (first row) is 10, cumulative 98 + 10=108. Stem 2 is 4, cumulative 108+4 = 112. Stem 3 is 2, cumulative 112+2 = 114. Stem 4 is 0, stem 5 is 1, stem 6 is 1. The 60th and 61st terms are in the stem 0 (second row) or stem 1 (second row)? Wait, actually, the median should be less than 50.

Step3: Check the maximum number of CDs

Looking at the histogram, the last bar is around 600 - 650? Wait, no. Wait the stem - and - leaf plot: the highest stem is 6 with leaf 0, but also, wait the stem - and - leaf plot's key is 0|1 represents 10? Wait, no, the key says "0|1 represents 10"? Wait, maybe I misread. Wait the stem - and - leaf plot: stem 6, leaf 0. If the key is, for example, stem is tens place and leaf is ones place? Wait no, the x - axis of the histogram is 0, 100, 200, 300, 400, 500, 600. So the stem - and - leaf plot's stem is probably hundreds or tens? Wait, no, the histogram has bins like 0 - 100, 100 - 200, etc. The stem - and - leaf plot: let's recalculate the counts. The first row: stem 6, leaf 0, count 1. Stem 5, leaf 0, count 1. Stem 4, no leaf, count 0. Stem 3, leaf 00, count 2. Stem 2, leaf 5555, count 4. Stem 1, leaf 55555556, count 10. Stem 1, leaf 0000000000000033, count 17. Stem 0, leaf 5555555555555555555566667889, count 29. Stem 0, leaf 1111111111111222222223333333333333333333344444444, count 52. Now, the maximum number of CDs: the highest stem is 6, but the histogram's last bin is around 600 - 650? Wait no, the stem - and - leaf plot's stem 6, leaf 0: if the stem is the hundreds place, then 6|0 is 60, but that doesn't match the histogram. Wait, maybe the key is 0|1 represents 10, so stem is tens and leaf is ones. Then stem 6, leaf 0 is 60, but the histogram has a bin up to 600. So there is a mistake in my initial analysis. Wait, the correct way: the histogram shows that the maximum number of CDs is not between 600 and 650. Because the stem - and - leaf plot's highest stem with non - zero count is 6, but if we look at the histogram, the last bar is very small, but the x - axis goes up to 600. Wait, no, the option says "The maximum number of CDs owned is between 600 and 650". But from the stem - and - leaf plot, the highest value (from the stem - and - leaf plot's last row) is stem 6, leaf 0, but if the stem is, say, hundreds, then 6|0 is 600? Wait, maybe the key is 0|1 represents 10, so stem is hundreds, leaf is tens? No, this is confusing. Wait, let's check the other options:
- Option 1: Median less than 50. Since the distribution is right - skewed and most of the data is on the left, median is less than 50. Correct.
- Option 3: Mean > median for right - skewed. Correct.
- Option 4: Distribution is right - skewed (tail on the right). Correct.
- Option 5: Number of students with less than 150 CDs: Let's sum the counts. Stem 0 (second row):52, stem 0 (first row):29, stem 1 (second row):17, stem 1 (first row):10, stem 2:4, stem 3:2, stem 4:0, stem 5:1, stem 6:1. Wait, no, to get less than 150: Let's see, if the stem is tens, then 0 - 149. But maybe the correct way is that the cumulative count up to a certain point. Wait, the option says 98 students own less than 150. Let's check the counts: 52 (stem 0, second row) + 29 (stem 0, first row)+17 (stem 1, second row) = 52 + 29+17=98. So this is correct.
- Option 2: The maximum number of CDs owned is between 600 and 650. But from the stem - and - leaf plot, the highest stem is 6, but if we look at the histogram, the last bar is at 600, but the stem - and - leaf plot's stem 6 has count 1, but maybe the value is 60 (if stem is tens) or 600 (if stem is hundreds). But the histogram's x - axis is 0, 100, 200, 300, 400, 500, 600. So the maximum is not between 600 and 650. So this statement does not describe the distribution.

Answer:

The maximum number of CDs owned is between 600 and 650.