Math 155: Worksheet, December 8, 2011.
Name:
1. A Chi-Square exercise from 11.1. Do the preferred methods of remote communication differ in
the age groups 15–24 and 25–34? Random samples of 100 people from each age group were taken
with the following results on preferred method of communication.
Age
Cell Phone
15–24
48
25–34
41
Column Total
89
Instant Message
40
30
70
E-mail
5
15
20
Other
7
14
21
Row Total
100
100
200
Conduct the hypothesis test: H0 : The proportions of the different age groups having each communication preference are the same; H1 : The proportions of the different age groups having each
communication preference are not (all) the same. Use a level of significance of α = .05.
Use the combined totals to give the hypothesized (equal) population proportion. For example, the
proportion preferring Cell Phone is 89/200 = .445. Then use this to compute the expected number
in each category for each population. For example, for the age 15–24 group, the expected prefering
cell phone is (.445)(100) = 44.5.
(a) For the instant messaging category, compute E for each age group, then compute
each age group in the instant messaging category.
Age Group 15 – 24 (IM category): E =
,
(O − E)2
=
E
Age Group 25 – 34 (IM category): E =
,
(O − E)2
=
E
(b) As in 11.2, χ2 =
this case
P (O−E)2
E
(O − E)2
for
E
and we sum this over each age group (population) and category, in
3.52
3.52
52
52
52
52
3.52
3.52
+
+
+
+
+
+
+
≈ 9.312
44.5 44.5 35 35 10 10 10.5 10.5
The number of degrees of freedom is the (n − 1)(m − 1) where n is the number of populations and
m is the number of categories. In this case
χ2 =
d.f. =
(c) Compute the P -value for the test, and determine whether or not to reject H0 .
2. Nationally about 28% of the population believes that NAFTA benefits America. A random
sample of 48 interstate truck drivers showed that 19 believe NAFTA benefits America. Conduct an
hypothesis test to determine whether the population proportion of interstate truckers who believe
NAFTA benefits America is higher than 28%. Test at a level of significance of α = .05.
(a) State the null and alternative hypotheses. Is this a right-tailed, left-tailed or two-tailed test?
(b) Find the P-value for the test, and report your conclusion for the test.
(c) Would you reject the null hypothesis at a level of significance of α = .04?
(d) Would you reject the null hypothesis at a level of significance of α = .01?
(e) Do you think the proportion of interstate truckers who believe NAFTA benefits America is higher
than 28%? Explain your answer.
3. The following sample data concerns the number of years a student studied German in school
versus their score on a proficiency test.
Years (x)
3
4
4
2
5
3
4 5
3
2
Test Score (y) 57 78 72 58 89 63 73 84 75 48
P
P
P 2
P 2
P
The sums are:
x = 35
y = 697
x = 133
y = 50085
xy = 2554
(a) Find the equation of the least squares line for this data.
(b) Use your line from (a) to predict the score on the proficiency test of a person who had 3.5 years
of German.
(c) Use the regression line in (a) to predict the number of years of German required to achieve a
proficiency score of 75.
(d) Compute the correlation coefficient r for this data. What does this coefficient suggest about a
linear relationship between number of years German was studied in school and test scores for this
sample? That is, determine whether it is a good fit, and whether it indicates a positive or negative
linear relationship.
(e) Compute the coefficient of determination, and interpret what it means.
4. (a) A local radio station claims that 15 percent of all people in Riverside say it is their favorite
station, 65 percent of all people in Riverside listen to it occasionally, while 20 percent never listen
to it. Suppose you surveyed 200 randomly selected people in Riverside and found that of those 200
people, 20 claimed it was their favorite station, 131 said they listen to it occasionally, while 49 never
listen to it. Conduct an hypothesis test at a level of significance of .05 to determine whether the
station’s claim concerning the distribution of listeners is correct.
(b) Explain why the sample size was sufficiently large for this test to be valid.