a Start by drawing a number line that includes the least and greatest value of each data set. Then graph points for the five-number summary of the data.
Freshmen's Distribution: Symmetric Sophomores' Distribution: Skewed left
B
b See solution.
C
c About 31 of the Freshmen surveyed
D
d About 95 of the Freshmen surveyed
Practice makes perfect
a We will begin by looking at the table that shows the results of a survey that asked freshmen and sophomores how many songs they have downloaded on their MP3 players.
Freshmen
Sophomores
Survey size
45
54
Minimum
250
360
Maximum
2150
2400
1^(st) Quartile
800
780
Median
1200
2000
3^(rd) Quartile
1600
2200
Mean
1150
1650
Standard Deviation
420
480
We will make a double box-and-whisker plot that represents the data and describe each distribution. Let's first draw a number line that includes the least and greatest value of each data set. Then we will graph points above the number line for the five-number summary.
Now, let's draw the box for each plot by using Q_1 and Q_3. Then we will draw a line through the median and the whiskers from the box to the least and greatest values of each data set.
We can see that for freshmen the whisker lengths are equal and the median is in the middle of the plot. Therefore, its distribution is symmetric. For sophomores, the whisker and and box to the left of the median are larger than the whisker and box to the right of the median. Therefore, its distribution is skewed left.
b In this part we will compare the number of songs downloaded for freshmen and sophomores. Let's first look at the median and the mean of both data sets.
&Freshmen&Sophomores
Median:&1200&2000
Mean:&1150&1650
We can see that the median for sophomores is almost twice the median for freshmen. Additionally, sophomores mean is significantly higher than freshmen mean. Now, let's look at the standard deviations.
Freshmen&Sophomores
420&480
The standard deviation for sophomores is greater too, so we can say that the centers and spread of the two data sets are quite different. Therefore, we can conclude that the number of songs downloaded by freshmen is less than those for sophomores. Additionally, sophomores have more variability in the number of songs downloaded.
c Assuming the symmetric distribution is bell-shaped, we know about 68 % of the data lies between 1 standard deviation of the mean. Because the mean is 1150 and the standard deviation is 420, we can see that 730- 1570 represents 68 % of the data.
1150- 420&= 730songs
1150+ 420&= 1570songs
Therefore, we would expect about 68 % of the freshmen surveyed to have between 730 and 1560 songs. Let's multiply 0.68 by 45 to find this value.
0.68*45=30.6
Rounging this value up, we can state that about 31 freshmen would have between 730 and 1560 songs downloaded on their MP3 players.
d Assuming the symmetric distribution is bell-shaped, we know about 95 % of the data lie between 2 standard deviations of the mean. Because the mean is 1150 and the standard deviation is 420, we can see that 310- 1990 represents 95 % of the data
Interval:
1150-2(420)&= 310 songs
1150+2(420)&= 1570songs
Therefore, we would expect about 95 % of the 100 freshmen surveyed to have between 310 and 1990 songs. Let's multiply 0.95 by 100 to find this value.
0.95*100=95
We can state that about 95 of the freshmen would have between 310 and 1570 songs downloaded on their MP3 players.
