Name: ______________________ Class: _________________ Date: _________
ID: A
Exam - Chapter 9
Free Response - use the significance test worksheet to complete each problem. 20 points each
1. Eleven percent of the products produced by an industrial process over the past several months have
failed to conform to specifications. The company modifies the process in an attempt to reduce the rate
of nonconformities. In a random sample of 300 items from a trial run, the modified process produces 16
nonconforming items. Do these results provide convincing evidence that the modification is effective?
Support your conclusion with a test of significance. Use a confidence interval to provide additional
support. Use α =.05.
2. Does too much sleep impair intellectual performance? Researchers examined this commonly held belief
by comparing the performance of subjects on the mornings following (a) two normal night’s sleep and
(b) two nights of “extended sleep.” The order of these two treatments was determined randomly. In the
morning they were given a number of tests of ability to think quickly and clearly. One test was for
vigilance where the lower the score, the more vigilant the subject, (vigilance = alertness). The following
data were collected:
Subject
Normal Sleep
Extended Sleep
A
8
8
B
9
9
C
14
17
Vigilance Score
D
E
F
G
4
12
11
3
2
21
16
9
H
26
38
I
3
10
J
8
0
Carry out an appropriate test to help answer the researchers’ question. (Hint: You want to know if the
extended sleep - normal sleep is ≠ 0). Use α =.05.
Multiple Choice - 6 points each
Identify the choice that best completes the statement or answers the question.
____
1. Which of the following increases the power of a significance test?
A. Perform the test many times using the same data.
B. Increasing the significance level .
C. Finding a way to increase the population standard deviation .
D. Using a two-tailed test instead of a one-tailed test.
E. Decreasing the size of your sample.
1
Name: ______________________
____
ID: A
2. The recommended daily Calcium intake for women over 21 (and under 50) is 1000 mg per day. The
health services at a college are concerned that women at the college get less Calcium than that, so they
take a random sample of female students in order to test the hypotheses
versus
. Prior to the study they estimate that the power of their test against the alternative
is 0.85. Which of the following is the best interpretation of this value?
A. The probability
1000.
B. The probability
C. The probability
D. The probability
900.
E. The probability
of failing to reject the null hypothesis when the parameter value is
of making a Type II error.
of rejecting the null hypothesis when the parameter value is 1000.
of failing to reject the null hypothesis when the parameter value is
of rejecting the null hypothesis when the parameter value is 900.
____
3. Which of the following statements is/are correct?
I. The power of a significance test depends on the alternative value of the parameter.
II. The probability of a Type II error is equal to the significance level of the test.
III. As α increases β decreases.
A. I, II, and III
B. II and III only
C. I and II only
D. I and III only
E. None of the above gives the complete set of correct responses.
____
4. A significance test was performed to test the null hypothesis H0 : µ = 2 versus the alternative
Ha: µ
2. A sample of size 28 produced a test statistic is t = 2.051. Assuming all conditions for
inference were met, which of the following intervals contains the P-value for this test?
A.
B.
C.
D.
E.
2
Name: ______________________
____
ID: A
5. Bags of a certain brand of tortilla chips claim to have a net weight of 14 ounces. Net weights actually
vary slightly from bag to bag and are Normally distributed with mean . A representative of a consumer
advocacy group wishes to see if there is any evidence that the mean net weight is less than advertised and
so intends to test the hypotheses
A Type I error in this situation would mean
A. concluding that the bags are not being underfilled when they actually aren’t.
B. concluding that the bags are being underfilled when they actually aren’t.
C. concluding that the bags are being underfilled when they actually are.
D. concluding that the bags are not being underfilled when they actually are.
E. none of these
____
6. In a test of H0 : p = 0.7 against Ha : p
0.7, a sample of size 80 produces z = 0.8 for the value of the test
statistic. Which of the following is closest to the P-value of the test?
A. 0.4238
B. 0.4681
C. 0.2090
D. 0.7881
E. 0.2119
____
7. The mean time it takes for a person to experience pain relief from aspirin is 25 minutes. A new
ingredient is added to help speed up relief. Let µ denote the mean time to obtain pain relief with the new
product. An experiment is conducted to verify if the new product works more quickly. What are the null
and alternative hypotheses for the appropriate test of significance?
A. H0 : µ = 25 vs. Ha : µ 25
B. H0 : µ = 25 vs. Ha : µ > 25
C. H0 : µ = 25 vs. Ha : µ < 25
D. H0 : µ < 25 vs. Ha : µ = 25
E. H0 : µ < 25 vs. Ha : µ > 25
____
8. Because t procedures are robust, the most important condition for their use is
A. the population distribution is approximately Normal.
B. the population standard deviation is known.
C. the data can be regarded as a random from the population.
D. np and n(1 – p) are both at least 10.
E. all values in the sample are within two standard deviations of the mean.
3
Name: ______________________
____
ID: A
9. A P-value indicates
A. the probability that the null
B.
hypothesis is true.
the probability of the observed statistic
given that the null hypothesis is true.
D. the probability the null is true given the
observed statistic.
E. the probability of the observed statistic
given that the alternative hypothesis is
true.
C. the probability that the alternative
hypothesis is true.
____ 10. Some people say that more babies are born in September than in any other month. To test this claim,
you take a simple random sample of 150 students at your school and find that 21 of them were born in
September. You are interested in whether the proportion born in September is higher than 1/12—what
you would expect if September was no different from any other month. Thus your null hypothesis is
. The P-value for your test is 0.0056. Which of the following statements best describes
what the P-value measures?
A. The probability that September birthdays are no more common that any other
month is 0.0056.
B. 0.0056 is the probability of getting a sample with a proportion of September
birthdays this close to 1/12 if the true proportion is not 1/12.
C. The probability that the proportion of September birthdays in the population is not
equal to 1/12 is 0.0056.
D. The probability that September birthdays are more common is 0.0056.
E. 0.0056 is the probability of getting a sample with a proportion of September
birthdays this far or farther above 1/12 if the true proportion is 1/12.
4
ID: A
Exam - Chapter 9
Answer Section
SHORT ANSWER
1. ANS:
State: We wish to test
, where p = the true proportion of
nonconforming items. We will use a significance level of α = 0.05. Plan: The procedure is a
one-sample z-test for a proportion. Conditions: Random: A simple random sample of items was taken.
Normal:
Independent: We must
assume that whether one item conforms to specifications has no impact on whether the next one
produced by the process conforms to specifications, and that the trial run consisted of at least
items.
Do:
, so
; P-value = 0.0008. Conclude: A P-value of
0.0008 is less than α = 0.05, so we reject H0 and conclude that there is convincing evidence that the true
proportion of nonconforming items is less than 0.11.
PTS: 1
2. ANS:
REF: Quiz 9.2A
TOP: Tests about a Population Proportion
State: We wish to test H 0 :µ D = 0, H A :µ D ≠ 0, where
= the mean difference in vigilance score
(extended sleep – normal sleep) for all possible subject of the experiment. Since lower values indicate
more vigilance, we are looking for positive values of this difference. We will use a significance level
of α = 0.05. Plan: The procedure is a one-sample t-test on paired data. Conditions: Random: The
treatments (extended and normal sleep) were assigned in random order. Normal: The boxplot (or
histogram) does not indicate strong skewness, and the extreme values are not outliers by the 1.5 x
IQR standard. Independent: It is reasonable to assume that the vigilance differences from subject to
subject are independent. Do:x = 3.2 ;t = 3.2 − 0 ≈ 1.73 ; df = 9; P-value = 0.119. Conclude: A
5.87
10
S x = 5.87
P-value of 0.119 is greater than α = 0.05, so we fail to reject H0; we do not have convincing evidence
that extended sleep reduces vigilance.
PTS: 1
REF: Quiz 9.3C
TOP: Tests about a Population Mean
1
ID: A
MULTIPLE CHOICE
1. ANS: B
/D/Correct! Raising α increases the probability of a Type I error but decreases the probability of a Type
II error, thereby increasing power.
PTS: 1
REF: Test 9A
2. ANS: E
/C/Correct! Power is the probability of rejecting a false null hypothesis, which in this case is concluding
that the true mean daily Calcium intake for women a the college is lower than 1000 mg when it is
actually 900 mg.
PTS: 1
REF: Test 9B
3. ANS: D
/B/Correct! I is true because power is greater if the alternative (actual) value of the parameter is farther
from the null. III is true because P(Type I error) = α and P(Type II error) depends on α. II is false,
since P(Type I error) = α, not P(Type II error).
PTS: 1
REF: Test 9B
4. ANS: A
/E/Correct! t = 2.051 is between 0.025 and 0.05 on the t-table for 27 degrees of freedom. Since the test
is two-tailed, we double the P-value, so it is between 0.05 and 0.10.
PTS: 1
REF: Test 9C
5. ANS: B
/A/Correct! Type I error is rejecting H0 when it’s true, thus concluding that µ is below 14 ounces when it
is not.
PTS: 1
REF: Test 9A
6. ANS: A
/C/Correct! Area under standard normal curve above z = 0.8 is 0.2119. Since the test is two-tailed, the
P-value is
PTS: 1
REF: Test 9A
7. ANS: C
/B/Correct! The question is whether the new product works more quickly, so we are interested in the
one-tailed alternative for the mean time for pain relief is less than 25 minutes.
PTS: 1
REF: Test 9B
8. ANS: C
/C/Correct! t-procedures are robust with respect to the Normality of the population if n is large enough,
so the most important condition is randomness.
PTS: 1
9. ANS: B
REF: Test 9B
PTS: 1
2
ID: A
10. ANS: E
/D/Correct! The P-value measures the probability of getting a result as extreme as the sample statistic is
when the null hypothesis is true.
PTS: 1
REF: Test 9C
3