Download Homework 2.5

Math For Liberal Arts I
Solutions to Homework: Section 3.1, pages 65-66, #1, 2, 4, 6, 8
#1, page 65. In the example described in the text, what percentage of the data is within
one standard deviation from the mean? Within two standard deviations?
SOLUTION: First we know that the mean is 294.625, and standard deviation is 142.
The interval within one standard deviation from the mean is from 152.625 to 436.625,
and the interval within two standard deviations from the mean is from 10.625 to 578.625.
There are 19 numbers from the data follow into the interval within one standard deviation
from the mean, hence 19/24=79.16% of the data is within one standard deviation from the
mean. Similarly, 24 of the 24 data follow in two standard deviations from the mean,
hence 100% of the data is within two standard deviations from the mean.
#2, page 65. Make a second histogram for the example described in text, grouping the
numbers from 0 to 49, 50 to 99, 100 to 149, and so on. Make a third histogram for that
example, grouping the numbers from 0 to 199, 200 to 399, 400 to 599, and 600 to 799.
SOLUTION: Grouping into 0-49, 50-99, 100-149, 150-199, 200-249, 250-299, 300349, 350-399, 400-449,
450-499, 500-549, 550-599, 600-649.
Histogram of the number of pages
7
6
Frequency
5
4
3
2
1
0
0
100
200
300
pages
400
500
600
Grouping into 0-199, 200-399, 400-599 and 600-799.
Histogram of the number of pages
14
12
Frequency
10
8
6
4
2
0
0
200
400
pages
600
800
#4, page 65. Out of 10 possible points, Ms. Smith’s class of 20 students made the
following test scores: 0, 1,3,3,4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10. Find the median,
mean and standard deviation of these test scores. What percentage of the scores lies
within one standard deviation from the mean? Within two standard deviations?
SOLUTION: The median is the middle number of the ordered data, which is between
10th and 11th, we take the average of 10th and 11th to obtain (6+6)/2=6, the mean is
average of 20 numbers giving 6 too. The standard deviation is the squared root of the
average of squared deviations giving 2.7386 or square root of 7.5. So the interval within
one standard deviation from the mean is from 5.26 to 8.74 and within two standard
deviations is from .52 to 11.48. Hence there are 8 out 20 within one standard deviation,
and 20 out of 20 within two standard deviations.
#6, page 65. What can you say about a set of numbers if their standard deviation is zero?
SOLUTION: All numbers are same.
#8, page 66. (i) Find five numbers whose median is greater than their mean. (ii) Find
five numbers whose median is less than their mean.(iii) Find five numbers whose median
equals their mean.
SOLUTION: (i) {3 2 3 4 5} gives median 3 and mean 2.4 hence median is greater than
mean.
(ii) {1 2 3 5 9} gives median=3 and mean= 4, hence median is less than mean
(iii) {1 2 3 4 5} gives both median and mean are equal to 3.
Solutions to Homework: Section 3.2, pages 69, #1, 2, 3, 4
#1, page 69. In the Swingsville scene, where we know that there are 300 Republicans
and 200 Democrats, what is the probability that a random sample of 50 will contain
exactly 30 Republicans and 20 Democrats?
SOLUTION: 300C30 • 200C20/ 500C50.
#2, page 69. Suppose Smallsville has only 50 registered voters: 30 Republicans and 20
Democrats. If a random sample of five is chosen, what is the probability that three are
Republicans and two are Democrats?
SOLUTION: 30C3 • 20C2/ 50C5.
# 3, page 69. Look at Problem 2 again, and find the probability of all possible political
makeups of a sample of five voters. Partial answers (to the nearest two decimal places)
are
Probability of 5R amd 0D=0.07
Probability of 4R amd 1D=0.26
Probability of 3R amd 2D=0.36
SOLUTION: Compute the rest of probabilities:
Probability of 2R amd 3D=30C2 • 20C3/ 50C5=27550/2118760=0.234052=0.23
Probability of 1R amd 4D=30C1 • 20C4/ 50C5=145350/2118760=0.068601=0.07
Probability of 0R amd 5D=30C0 • 20C5/ 50C5=15504/2118760=0.007317=0.01
Notice that the sum of the probabilities is equal to one exactly.
#4, page 69. Given again the case that a sample of 50 is chosen from the unknown
political makeup of Dullsville with population 500 people and that in the sample there are
30 Republicans and 20 Democrats, what is the only sure conclusion you can make about
the number of Republicans in Dullsville?
SOLUTION: We are sure that in the sample, the proportion of Republicans is
30/50=60% in the sample. Therefore, we can use this proportion as an estimate of the
population proportion of Republicans. Hence we can estimate that there is about
60%*500=300 Republicans in Dullsville.