1. A hypothesis test returns a p-value of 0.06 against the null hypothesis: The identification of an
elementary student as"gifted" is independent of their identification as dyslexic. What is a
correct interpretation of this p-value?
a) Only 6% of dyslexic students are identified as "gifted".
b) Elementary students with dyslexia are only identified as "gifted" 96% as often as their nondyslexic peers.
c) There is a 94% chance that dyslexia is independent from whether an elementary student is
identified as "gifted".
d) If dyslexia is independent from whether an elementary student is identified as
"gifted", we would get a result indicating at least this much connection between the
two 6% of the time.
e) None of these is a correct interpretation of the p value.
2. A sales manager would like to know whether the male or female employees at her company are
more likely to have a larger average number of sales. The manager does not know either
population standard deviation. Which test statistic would be most appropriate to use in this
hypothesis test?
k
( f ijβe ij ) 2 2
( f iβe i )2 2
2
2
:Ο Ο β₯ΟΞ± ,(rowsβ1)(col β1 )
:Ο Ο β₯ΟΞ± ,(rowsβ1)
a) Ο Ο =βi β j
b) Ο Ο =β
eij
ei
i=1
( pΜ β pΜ2)β( p1β p2 )
( xΜ β xΜ )β(ΞΌ 1βΞΌ 2)
( xΜ β xΜ )β(ΞΌ1βΞΌ 2 )
c) z= 1
d) z= 1 2
e) t= 1 2
x
2
2
2
2
1 1
Ο
Ο
s
s
Μp (1β Μp )( + )
( 1+ 2)
( 1+ 2)
n1 n 2
n1 n2
n1 n2
β
β
β
3. A school district equity officer wants to test the null hypothesis that dyslexia is independent of
whether an elementary student is identified as "gifted". Which test statistic would be most
appropriate to use in the hypothesis test?
k
( f ijβe ij ) 2 2
( f iβe i )2 2
2
2
:Ο Ο β₯ΟΞ± ,(rowsβ1)(col β1 ) X
:Ο Ο β₯ΟΞ± ,(rowsβ1)
a) Ο Ο =βi β j
b) Ο Ο =β
eij
ei
i=1
( pΜ β pΜ2)β( p1β p2 )
( xΜ β xΜ )β(ΞΌ 1βΞΌ 2)
( xΜ β xΜ )β(ΞΌ1βΞΌ 2 )
c) z= 1
d) z= 1 2
e) t= 1 2
2
2
2
2
1 1
Ο1 Ο2
s1 s 2
Μp (1β Μp )( + )
( + )
( + )
n1 n 2
n n
n n
β
β
1
2
β
1
2
4. A politician wants to know whether the level of support for a certain ballot initiative is higher
among Republicans than Democrats. Which test statistic would be most appropriate to test this
hypothesis?
k
( f ijβe ij ) 2 2
( f iβe i )2 2
2
2
:Ο Ο β₯ΟΞ± ,(rowsβ1)(col β1 )
:Ο Ο β₯ΟΞ± ,(rowsβ1)
a) Ο Ο =βi β j
b) Ο Ο =β
eij
ei
i=1
( pΜ β pΜ2)β( p1β p2 )
( xΜ β xΜ )β(ΞΌ 1βΞΌ 2)
( xΜ β xΜ )β(ΞΌ1βΞΌ 2 )
c) z= 1
X d) z= 1 2
e) t= 1 2
1 1
Ο 21 Ο 22
s12 s 22
Μp (1β Μp )( + )
(
+
)
(
+ )
n1 n 2
n1 n2
n 1 n2
β
β
β
5. An amusement park believes that 25% of attendants are under 12 years old, 20% are between 12
and 16 years old, 15% are between 16 and 24 years old, and 40% are over 24 years old. Which
test statistic would be most appropriate to use in this hypothesis test?
k
( f ijβe ij ) 2 2
( f iβe i )2 2
2
2
:Ο Ο β₯ΟΞ± ,(rowsβ1)(col β1 )
:Ο Ο β₯ΟΞ± ,(rowsβ1) X
a) Ο Ο =βi β j
b) Ο Ο =β
eij
ei
i=1
( pΜ β pΜ2)β( p1β p2 )
( xΜ β xΜ )β(ΞΌ 1βΞΌ 2)
( xΜ β xΜ )β(ΞΌ1βΞΌ 2 )
c) z= 1
d) z= 1 2
e) t= 1 2
2
2
1 1
Ο1 Ο2
s12 s 22
Μp (1β Μp )( + )
( + )
( + )
n1 n 2
n1 n2
n1 n2
β
β
β
6. An electronics company wants to demonstrate that its devices have longer average battery life
than the next closest competitor. The two population standard deviations are assumed to be
known. Which test statistic would be most appropriate to use in this hypothesis test?
2
2
k
( f ijβe ij )
( f iβe i ) 2
2
2
:Ο 2Ο β₯ΟΞ± ,(rowsβ1)(col β1 )
:Ο Ο β₯ΟΞ± ,(rowsβ1)
a) Ο Ο =βi β j
b) Ο Ο =β
eij
ei
i=1
( pΜ β pΜ2)β( p1β p2 )
( xΜ β xΜ )β(ΞΌ1βΞΌ 2 )
( xΜ β xΜ )β(ΞΌ 1βΞΌ 2)
c) z= 1
d) t= 1 2
e) z= 1 2
X
2
2
2
2
1 1
s
s
Ο
Ο
Μp (1β Μp )( + )
( 1+ 2)
( 1+ 2)
n1 n 2
n1 n2
n1 n2
β
β
β
sd
to estimate the difference between means?
βn
When each element of the sample is measured twice (for instance "before" and "after"
data)
When the two underlying populations are not normally distributed.
When the two standard deviations are identical
When the two standard deviations are unknown
None of the above
7. When is it appropriate to use the equation dΜ Β±t
a)
b)
c)
d)
e)
8. A 95% confidence interval for the difference between two proportions is {.02 to .35}. What is
the result of a test at 90% confidence of the null hypothesis: there is no difference between the
two proportions?
a) Reject the null hypothesis for this data at the given level of significance
b) Fail to reject the null hypothesis for this data at the given level of significance
c) This problem cannot be answered because the two levels of confidence are different
d) This problem cannot be answered because there is no relationship between confidence
intervals and hypothesis tests.
e) None of the above
Use the following for questions 9 through 11: A study by the Pew Research Center for The People &
The Press showed that in July 2013, a minority of American adults approved of the NSA's surveillance
programs. However, when the question's wording mentioned that this surveillance happened with court
approval, substantially more people (37% instead of 25%) approved of the program.
9. What is the appropriate test statistic to check whether there was a statistically significant
difference between these two percentages, assuming there were 1000 people questioned in each
group?
(.37β.25)β0
a) z=
X
1
1
.31(1β.31)(
+
)
1000 1000
(.37β.25)β0
b) z=
1
1
.37(1β.37)(
+
)
1000 1000
(.37β.25)β0
c) z=
1
1
.37(1β.25)(
+
)
1000 1000
(.37β.25)β0
d) z=
1
1
.25(1β.25)(
+
)
1000 1000
e) none of the above.
β
β
β
β
10. What is the result of the test of the null hypothesis (Ξ±= 0.05) that mentioning court approval
has no effect on answers to this survey?
a) Because the p-value is large enough, we reject the null hypothesis
b) Because the p-value is small enough, we reject the null hypothesis
c) Because the p-value is too large, we cannot reject the null hypothesis
d) Because the p-value is too small, we cannot reject the null hypothesis
e) None of the above
11. Is it appropriate to test this question using a one-tailed or a two-tailed test?
a) One tailed because mentioning court approval could only possibly affect people's answers in
one direction.
b) Two tailed because it's possible that mentioning court approval could affect people's
answers in either direction.
c) One tailed because it's possible that mentioning court approval could affect people's answers
in either direction.
d) Two tailed because mentioning court approval could only possibly affect people's answers
in one direction.
e) none of the above
For question 12 and 13: The Gallup organization is interested in whether the Affordable
Care Act's ("Obamacare's") support may be changing. In August, Gallup reported that
among registered voters in the United States, 41% approved, 49% disapproved, and 11%
had no opinion (the percents do not add to 100% due to rounding error). In October, Gallup
surveyed 1528 registered voters: 679 approved, 762 disapproved, and 87 had no opinion.
Is there evidence that voters' opinions have changed between August and October?
12. What is the appropriate test statistic to test the null hypothesis- opinions have NOT changed
between August and October?
t=43.75
a)
b) Ο 2=43.75 X
c) t=0.027 d) Ο 2=0.027
13. What is the appropriate critical value for the hypothesis test above at 95% confidence?
a) 5.991
b) 7.378
c) 7.815
d) 9.348
e) none of these
For questions 14-18, consider the table below with opinions about the Affordable Care Act
broken down by age categories.
18 to 29
30 to 49
50 to 64
65+
Total
Approve
76
92
72
29
269
Disapprove
66
94
94
40
294
No Opinion
8
12
9
6
35
Total
150
200
175
75
600
14. How many 18-29 year-olds in this survey would be expected to disapprove of the Affordable
Care Act if age and opinion were independent?
a)
b)
c)
d)
e)
(66/150)*(66/294) = 0.098
(150/600)*(294/600)*600 = 73.5
66
(66/150)*(66/294)*600 = 59.27
None of the above
15. Can we use our chi-squared test for this data?
a)
b)
c)
d)
note that our "expected value"
No, because we expect a value of at least five in each box.
in one box is 4.375, calling th
No, because we expect a value of less than five in some boxes.
Yes, because we expect a value of at least 5 in each box chi-squared approach into quest
Yes, because we expect a value of less than five in some boxes.
16. Assuming we can use the chi-squared distribution for this test, how many degrees of freedom
would we have?
a) 3
b) 4
c) 6
d)12
e) none of these
17. If one of our assumptions is not met, which of the following is an option to still use the chisquared distribution?
a) Decrease the degrees of freedom for our chi-square statistic.
b) Separate columns or rows so that each box has an expected count less than five.
c) Use a z distribution instead
d) Combine columns or rows so that each box has at an expected count of at least five.
18. If we can perform a chi-squared distribution and we get a p-value of 0.002, what is an
appropriate statistical interpretation of that p-value?
a) Assuming the null hypothesis is correct, there is only a 0.2% chance of getting a result
this extreme. Such a low p-value is generally considered good evidence to reject the
null.
b) Assuming the alternative hypothesis is correct, there is only a 0.2% chance of getting a
result this extreme. Such a low p-value is generally considered good evidence to reject the
null.
c) Assuming the null hypothesis is correct, there is only a 0.2% chance of getting a result this
extreme. Such a large p-value is generally not generally considered enough evidence to
reject the null.
d) Assuming the alternative hypothesis is correct, there is only a 0.2% chance of getting a
result this extreme. Such a large p-value is generally not generally considered enough
evidence to reject the null.
e) none of these
19. Given the following information, calculate a 95% confidence interval for the difference
between the two population means ΞΌ1 and ΞΌ2:
s12=5 s 22=4 n1=17 n2 =15
4
5
5
4
b) β2Β±2.145
X
β2Β±2.145
+
+
17 15
17 15
xΜ1=6
a)
xΜ2=8
β
β
c) β2Β±1.96
5
4
+
17 15
d) β2Β±1.96
β
β
4
5
+
17 15
For questions 20-23, a company is interested in whether customer satisfaction depends on the sales region by
which they are served. The following table summarizes the data found by the researchers:
Region
Highly Satisfied
Somewhat Satisfied
Somewhat Dissatisfied Highly Dissatisfied
Total
Western
90
46
8
6
150
Southern
10
40
42
8
100
Midwestern
117
94
75
14
300
Eastern
171
144
65
20
400
26
10
2
50
Alaska & Hawaii 12
Total
400
350
200
50
1000
20. How do we calculate the table of expected values, assuming region and satisfaction are
independent? Based on the table below, calculate the expected number of customers in the
Western region who will be highly satisfied, assuming independence.
a) 60
Region
b) 30
Highly Satisfied
c) 900
Somewhat Satisfied
d)90
e) none of these
Somewhat Dissatisfied Highly Dissatisfied
Total
Western
150
Southern
100
Midwestern
300
Eastern
400
Alaska & Hawaii
50
Total
400
350
200
50
1000
21. When calculating the chi-squared statistic, what contribution is made by the difference in the
number of Alaskan or Hawaiian customers who are Somewhat Dissatisfied compared to the
expectation under the null?
a) 12.1
b) 10
c) 100
d)0
e) none of these
22. Unfortunately, this data cannot be tested using the chi-squared test statistic. Why?
a) The expected count in the bottom right box (Alaska & Hawaii highly dissatisfied) is too large.
b) The expected count in the bottom right box (Alaska & Hawaii highly dissatisfied) is too small.
c) The actual count in the bottom right box (Alaska & Hawaii highly dissatisfied) is too small.
d) The actual count in the bottom right box (Alaska & Hawaii highly dissatisfied) is too large.
e) None of these is a possible explanation.
23. What could be done to use the test after all?
a) Combine Alaska and Hawaii with another row (for instance Western Region)
b) Combine "Highly Dissatisfied" with another column (for instance Somewhat Dissatisfied)
c) Either (a) or (b) would work, although they might result in different p-values and
interpretations.
d) Either (a) or (b) would work, and they would result in identical p-values and interpretations
24. A statgraphics program returns the following output for a test of goodness of fit against the null
hypothesis - participants at a conference are evenly distributed across several responses to the
question "Which of the following was your strongest reason for attending the conference?":
Test
Chi-Square
Statistic
12.873
Df
7
P-Value
?????????
a) How many responses were there to the question in the study? (1 point)
8 (degrees if freedom = category -1)
b) Give an estimate for the p-value of this test. (1 point)
between 5% and 10%
c) What does this tell you about the participants at the conference and t
heir reasons for attending?(2 points)
Not all reasons for attending were equally likely to be selected.
(assuming alpha = 0.10; if alpha = 0.05 then we couldn't conclude anything)
25. A company wants to know whether their automotive fleet gets better gas economy (in terms of
miles per dollar) with regular or supreme gasoline.
a) Would you test this using a paired or unpaired t test? (1 point)
answers will vary
b) Why would you recommend this type of test?(3 points)
advantages to paired test: decrease number of cars needed, lower variability
because don't have to worry about variability between types of car
advantages to unpaired test: don't need to worry about order effects
26. Bonus: I'm interested in any feedback you're willing to give about how this class is going for
you so far. For one point each (optional, extra credit) please tell me:
a) What is one thing that has supported your learning in BA 376 so far this term?
b) What is one thing you could do better to get more out of this class?
c) What is one thing I could do better to help you get more out of this class?
Thank you!