Download Rayna Todorcheva 1154MATH330-002 Assignment Homework9

Rayna Todorcheva
Assignment Homework9-week 10 due 11/03/2015 at 12:15pm EST
1154MATH330-002
• B. pool the population proportions when the populations are normally distributed
• C. pool the population proportions when they are equal
• D. never pool the population proportions
Select True or False from each pull-down menu, depending
on whether the corresponding statement is true or false.
? 1. When the necessary conditions are met, a two-tail test is
conducted to test the difference between two population
proportions. The two sample proportions are p̂1 = 0.40
and p̂2 = 0.35 and the standard error of the sampling
distribution of p1 − p2 is 0.04. The calculated value of
the test statistic is 1.25.
? 2. In testing the difference between two population means
using two independent samples, the sampling distribution of the sample mean difference x̄1 − x̄2 is normal if
the sample sizes are both greater than 30.
? 3. A political analyst in Iowa surveys a random sample
of registered Democrats and compares the results with
those obtained from a random sample of registered Republicans. This is an example of two independent samples.
? 4. In testing for the equality of two population variances,
when the populations are normally distributed, the 5
Correct Answers:
• D
• D
4. (1 pt)
Two random samples are taken, with each group asked if they
support a particular candidate. A summary of the sample sizes
and proportions of each group answering “yes” are given below:
Pop. 1 : n1 = 93, p̂1 = 0.76
Pop. 2 : n2 = 92, p̂2 = 0.617
Suppose that the data yields (0.0164, 0.2696) for a confidence interval for the difference p1 − p2 of the population proportions. What is the confidence level? (Give your answer in
terms of percentages.)
Confidence Level =
Correct Answers:
• 94
Correct Answers:
• T
• F
• T
• F
5. (1 pt)
Two random samples are selected from two independent populations. A summary of the samples sizes, sample means, and
sample standard deviations is given below:
2. (1 pt) Samples are collected from two independent populations to conduct a hypothesis test for the difference of the means
µ1 − µ2 . If the sample size n2 from population 2 is 7 times larger
than the sample size n1 from population 1, and the degrees of
freedom for the test statistic is 158, then what are the sample
sizes?
n1 =
n2 =
n1 = 51, x̄1 = 51.7, s1 = 5.1
n2 = 44, x̄2 = 76.6, s2 = 10.8
Find a 95.5% confidence interval for the difference µ1 − µ2
of the means, assuming equal population variances.
Confidence Interval =
Correct Answers:
• (-28.3456183959344,-21.4543816040656)
Correct Answers:
• 20
• 140
6. (1 pt)
Two random samples are selected from two independent populations. A summary of the samples sizes and sample means is
given below:
3. (1 pt)
The ratio of two independent chi-squared variables divided
by their degrees of freedom is:
• A. Student t-distributed
• B. normally distributed
• C. chi-squared distributed
• D. F-distributed
In constructing a confidence interval estimate for the difference between two population proportions, we:
• A. pool the population proportions when the population
means are equal
n1 = 51, x̄1 = 55.1
n2 = 44, x̄2 = 76
If the 93% confidence interval for the difference µ1 − µ2 of
the means is (-24.0266, -17.7734), what is the value of the
pooled variance estimator? (You may assume equal population
variances.)
Pooled Variance Estimator =
Correct Answers:
• 68.7245161290323
1
7. (1 pt)
Random samples of resting heart rates are taken from two
groups. Population 1 exercises regularly, and Population 2 does
not. The data from these two samples is given below:
Population 1: 68, 65, 66, 72, 73, 71, 70
Population 2: 73, 69, 75, 72, 77, 68, 75, 74
Is there evidence, at an α = 0.035 level of significance, to
conclude that there those who exercise regularly have lower
resting heart rates? (Assume that the population variances are
equal.) Carry out an appropriate hypothesis test, filling in the
information requested.
A. The value of the standardized test statistic:
Note: For the next part, your answer should use interval notation. An answer of the form (−∞, a) is expressed (-infty, a), an
answer of the form (b, ∞) is expressed (b, infty), and an answer
of the form (−∞, a) ∪ (b, ∞) is expressed (-infty, a)U(b, infty).
B. The rejection region for the standardized test statistic:
Select True or False from each pull-down menu, depending
on whether the corresponding statement is true or false.
? 1. We say that two samples are dependent when the selection process for one is related to the selection process
for the other.
? 2. The pooled variances t−test requires that the two population variances are not the same.
? 3. In testing the difference between two population means
using two independent samples, we use the pooled variance in estimating the standard error of the sampling
distribution of the sample mean difference x̄1 − x̄2 if the
populations are normal with equal variances.
? 4. When comparing two population variances, we use the
ratio
rather than the difference σ21 − σ22 .
Correct Answers:
•
•
•
•
C. The p-value is
D. Your decision for the hypothesis test:
•
•
•
•
σ21
σ22
A. Reject H0 .
B. Do Not Reject H0 .
C. Reject H1 .
D. Do Not Reject H1 .
T
F
T
T
10. (1 pt)
Two statistics teachers both believe that each has the smarter
class. To put this to the test, they give the same final exam to
their students. A summary of the class sizes, class means, and
standard deviations is given below:
Correct Answers:
• -2.26104143398232
• (-infinity,-1.97416)
• 0.0207756
• A
n1 = 35, x̄1 = 84.8, s1 = 17.8
n2 = 46, x̄2 = 82.9, s2 = 17.5
Is there evidence, at an α = 0.055 level of significance, to
conclude that there is a difference in the two classes? (Assume
that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested.
A. The value of the standardized test statistic:
Note: For the next part, your answer should use interval notation. An answer of the form (−∞, a) is expressed (-infty, a), an
answer of the form (b, ∞) is expressed (b, infty), and an answer
of the form (−∞, a) ∪ (b, ∞) is expressed (-infty, a)U(b, infty).
B. The rejection region for the standardized test statistic:
8. (1 pt)
Two independent samples of sizes 25 and 35 are randomly
selected from two normal populations with equal variances. In
order to test difference between the population means, the test
statistic is:
• A. Student t distributed with 33 degrees of freedom
• B. a standard normal random variable
• C. Student t distributed with 58 degrees of freedom
• D. approximately standard normal random variable
Which of the following statements is not correct for an Fdistribution?
• A. Exact shape of the distribution is determined by two
numbers of degrees of freedom
• B. Degrees of freedom for the numerator can be larger,
smaller, or equal to the degrees of freedom for the denominator.
• C. Variables that are F-distributed range from 0 to 100
• D. Degrees of freedom for the denominator are always
smaller than the degrees of freedom for the numerator
C. The p-value is
D. Your decision for the hypothesis test:
•
•
•
•
A. Do Not Reject H0 .
B. Reject H1 .
C. Do Not Reject H1 .
D. Reject H0 .
Correct Answers:
•
•
•
•
Correct Answers:
• C
• D
2
0.480483114228723
(-infinity,-1.94773) U (1.94773,infinity)
0.632212
A
mean:
standard deviation:
b) Can you conclude that the variable x1 − x2 is approximately normally distributed? (Answer yes or no)
answer:
11. (1 pt) In order to compare the means of two populations, independent random samples of 33 observations are selected from each population, with the following results:
Sample 1 Sample 2
x1 = 1
x2 = 0
s1 = 180 s2 = 170
Correct Answers:
• -7
• 2.13623314820552
• yes
(a) Use a 96 % confidence interval to estimate the difference
between the population means (µ1 − µ2 ).
≤ (µ1 − µ2 ) ≤
(b) Test the null hypothesis: H0 : (µ1 − µ2 ) = 0 versus the alternative hypothesis: Ha : (µ1 − µ2 ) 6= 0. Using α = 0.04, give
the following:
(i) the test statistic z =
(ii) the positive critical z score
(iii) the negative critical z score
The final conclustion is
13. (1 pt) Suppose you want to use a paired sample to compare the mean TV viewing times of married men and married
women.
What is the variable under consideration?
• A. marital status
• B. TV viewing time
• C. mean TV viewing time
• D. None of the above
• A. We can reject the null hypothesis that (µ1 − µ2 ) = 0
and accept that (µ1 − µ2 ) 6= 0.
• B. There is not sufficient evidence to reject the null hypothesis that (µ1 − µ2 ) = 0.
What are the two populations under consideration?
• A. married people and single people
• B. men and women
• C. married men and married women
• D. None of the above
(c) Test the null hypothesis: H0 : (µ1 − µ2 ) = 24 versus the alternative hypothesis: Ha : (µ1 − µ2 ) 6= 24. Using α = 0.04, give
the following:
(i) the test statistic z =
(ii) the positive critical z score
(iii) the negative critical z score
The final conclustion is
What are the pairs?
• A. married couples
• B. men and women
• C. None of the above
• A. We can reject the null hypothesis that (µ1 − µ2 ) = 24
and accept that (µ1 − µ2 ) 6= 24.
• B. There is not sufficient evidence to reject the null hypothesis that (µ1 − µ2 ) = 24.
What is the paired-difference variable?
• A. the difference between the TV viewing times of a
married couple
• B. the difference between the TV viewing times of men
and women
• C. None of the above
Correct Answers:
•
•
•
•
•
•
•
•
•
•
Correct Answers:
• B
• C
• A
• A
-87.5158091565791
89.5158091565791
0.0232020699982196
2.05375
-2.05375
B
-0.533647609959052
2.05375
-2.05375
B
14. (1 pt) Industry Research polled teenagers on sunscreen
use. The survey revealed that 46% of teenage girls and 30%
of teenage boys regularly use sunscreen before going out in the
sun.
12. (1 pt) A variable of two populations has a mean of 41 and
a standard deviation of 10 for one of the populations and a mean
of 48 and a standard deviation of 10 for the other population.
identify the two populations
• A. teenage girls and teenage boys who use sunscreen
regularly
• B. teenage girls and teenage boys
• C. all teenagers
a) For independent samples of sizes 36 and 56, respectively,
find the mean and standard deviation of x1 − x2 .
3
• D. None of the above
16. (1 pt) Two polls were conducted about the upcoming
election for new governor. The first poll said that 45% of the
people asked would vote for Mr. Johnson with a margin of error
of plus or minus 4.6%. The second poll said that 42% of the
people asked would vote for Mr. Johnson with a margin of error
of plus or minus 3.5%.
identify the specified attribute
• A. uses sunscreen before going out in the sun
• B. being a teenage girl or a teenage boy
• C. being a teenager
• D. None of the above
Can the conclusions of both polls be correct?
• A. no, the difference between 45% and 42% is too large.
• B. yes, the two corresponding confidence intervals
overlap.
• C. no, the two corresponding confidence intervals do
not overlap.
are the proportions 0.46 (46%) and 0.30 (30%) population
proportions or a sample proportions?
• A. sample proportions
• B. population proportions
• C. None of the above
Correct Answers:
• B
17. (1 pt) Consider two normal distributions, one with mean
-4 and standard deviation 3, the other with mean 6 and standard
deviation 3. Answer the following statements using true or false.
Correct Answers:
• B
• A
• A
a) The two distributions have the same shape.
answer:
15. (1 pt) In order to compare the means of two populations,
independent random samples of 402 observations are selected
from each population, with the following results:
Sample 1
x1 = 5476
s1 = 190
b) The two distributions are centered at the same place.
answer:
Sample 2
x2 = 5044
s2 = 195
Correct Answers:
• true
• false
(a) Use a 95 % confidence interval to estimate the difference
between the population means (µ1 − µ2 ).
≤ (µ1 − µ2 ) ≤
(b) Test the null hypothesis: H0 : µ1 = µ2 versus the alternative
hypothesis: Ha : µ1 6= µ2 . Using α = 0.05, give the following:
(i) the test statistic t =
(ii) the positive critical t score
(iii) the negative critical t score
The final conclustion is
• A. We can reject the null hypothesis that µ1 = µ2 and
conclude that µ1 6= µ2 .
• B. There is not sufficient evidence to reject the null hypothesis that µ1 = µ2
18. (1 pt)
The sample size needed to estimate the difference between
two population proportions to within a margin of error E with a
significance level of α can be found as follows. In the expression
r
p1 q1 p2 q2
E = zα/2
+
n1
n2
we replace both n1 and n2 by n (assuming that both samples
have the same size) and replace each of p1 , p2 , q1 , and q2 by
0.5 (because their values are not known). Then we solve for n,
and get
(zα/2 )2
.
n=
2E 2
Finally, increase the value of n to the next larger integer number.
Use the above formula to find the size of each sample needed
to estimate the difference between the proportions of boys and
girls under 10 years old who are afraid of spiders. Assume that
we want 95% confidence that the error is smaller than 0.04.
n=
Correct Answers:
• 405.345279115845
• 458.654720884155
• 31.8137174906259
• 1.96293
• -1.96293
• A
4
Correct Answers:
• 1201
(g) Construct a 95% conficence interval for the population
mean of all differences x − y.
19. (1 pt) Test the claim that the two samples described below come from populations with the same mean. Assume that
the samples are independent simple random samples. Use a
significance level of 0.01.
Sample 1: n1 = 78, x1 = 19, s1 = 1.
Sample 2: n2 = 72, x2 = 20, s2 = 5.
< µd <
Correct Answers:
•
•
•
•
•
•
•
•
The test statistic is
The P-Value is
The conclusion is
• A. There is not sufficient evidence to warrant rejection
of the claim that the two populations have the same
mean.
• B. There is sufficient evidence to warrant rejection of
the claim that the two populations have the same mean.
21. (1 pt) The purpose of this question is to compare the
variability of x1 and x2 with the variability of (x1 − x2 ).
(a) Suppose the first sample of 100 observations is selected
from a population with mean µ1 = 200 and variance σ21 = 860.
Construct an interval extending 2 standard deviations of x1 on
each side of µ1 .
≤ µ1 ≤
(b) Suppose the second sample of 100 observations is selected from a population with mean µ2 = 200 and variance
σ22 = 830. Construct an interval extending 2 standard deviations
of x2 on each side of µ2 .
≤ µ2 ≤
(c) Consider the difference between the two sample means
(x1 − x2 ). Compute the mean and the standard deviation of the
sampling distribution of (x1 − x2 ).
mean =
standard deviation =
(d) Based on 100 observations, construct an interval extending 2 standard deviations of (x1 − x2 ) on each side of (µ1 − µ2 )
≤ (µ1 − µ2 ) ≤
Correct Answers:
• -1.66656775176307
• 0.0956004
• A
20. (1 pt) Suppose you want to test the claim the the paired
sample data given below come from a population for which the
mean difference is µd = 0.
x
y
85
73
78
81
50
61
87
73
66
59
60
86
2.14285714285714
16.5673229725321
0.34220779691071
2.44690
-2.4469
A
-13.1792868590963
17.4650011448106
90
68
Use a 0.05 significance level to find the following:
(a) The mean value of the differnces d for the paired sample
data
d=
Correct Answers:
•
•
•
•
•
•
•
•
(b) The standard deviation of the differences d for the paired
sample data
sd =
(c) The t test statistic
t=
(d) The positive critical value
t=
194.134848680554
205.865151319446
194.238055883645
205.761944116355
0
4.11096095821889
-8.22192191643779
8.22192191643779
22. (1 pt) Two independent samples have been selected, 87
observations from population 1 and 70 observations from population 2. The sample means have been calculated to be x1 = 12
and x2 = 10.7. From previous experience with these populations, it is known that the variances are σ21 = 24 and σ22 = 38.
(a) Find σ(x1 −x2 ) .
answer:
(b) Determine the rejection region for the test of H0 :
(µ1 − µ2 ) = 3.46 and Ha : (µ1 − µ2 ) > 3.46 Use α = 0.02.
z>
(e) The negative critical value
t=
(f) Does the test statistic fall in the critical region?
• A. No
• B. Yes
5
(c) Compute the test statistic.
z=
The final conclustion is
• A. There is not sufficient evidence to reject the null hypothesis that (µ1 − µ2 ) = 3.46.
• B. We can reject the null hypothesis that (µ1 − µ2 ) =
3.46 and accept that (µ1 − µ2 ) > 3.46.
(d) Construct a 98 % confidence interval for (µ1 − µ2 ).
≤ (µ1 − µ2 ) ≤
• 3.40495369102906
23. (1 pt) A paired difference experiment yielded nD pairs of
observations. In each case described below, what is the rejection
region for testing H0 : µ = 2 against Ha : µ > 2? Use sD = 12.5.
(a) nD = 43, α = 0.07
z>
(b) nD = 29, α = 0.02
t>
(c) nD = 21, α = 0.06
t>
Correct Answers:
• 0.904831040483614
• 2.05375
• -2.38718600861164
• A
• -0.804953691029056
Correct Answers:
• 1.47579
• 2.15393
• 1.62415
c
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