Math 244 Excel Assignment 5 (EA5)
Chapter 13: Hypothesis Testing for Means
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Quest 1 Radon detectors are manufactured for home use by the Brand X corporation. Researchers
are concerned that they may be inaccurate. They place 24 randomly selected Brand X radon
detectors in a chamber where they are exposed to radiation at a level of precisely
105 picocuries per liter of radon. The readings are given to the right, in picocuries per liter
of radon. At a 5% significance level, do the sample results indicate that the mean reading
for Brand X detectors is incorrect?
(A) Write the hypotheses for this test.
H0
Ha
(B) State the conditions needed for the test you are conducting, and verify that the conditions are satisfied.
Insert extra lines as needed.
Click here for a video showing how to insert blank lines.
Click here for a video showing how to check conditions for these kinds of tests.
Condition 1:
Condition 2:
(C) Use Excel formulas to compute the following.
Click here for a video showing how to find the test statistic and p-value for these kinds of tests.
sample estimate x̄
null value μ0
sample st dev s
sample size n
null std error
test statistic t
df
p-value
(D) Write a sentence or two explaining what your p-value means in the context of the problem.
(E) Is the sample result statistically significant at the 5% level? Write a sentence explaining what numerical
comparison leads to your answer.
Significant (yes/no)?
Explanation:
(F) Write a conclusion in the context of the problem.
Readings
97.4 101.1 103.5 105.3
97.9 102 103.6 107.4
98.6 102.7 103.7 108.6
99.6 102.9 103.8 109.2
100.3 103.1 104.2 109.3
100.4 103.3 104.9 109.6
nditions are satisfied.
nds of tests.
ining what numerical
Math 244 Excel Assignment 5 (EA5)
Hypothesis Testing for Means
Page 2
Click here to see all Math 244 Excel assignments and videos.
Practice questions are available on the last tab of this workbook.
Quest 2 The developers of a training program designed to improve manual dexterity claim that people who
complete the 6-week program will increase their manual dexterity. A random sample of 12 people
was selected from those enrolled in the training program. A measure of each person’s dexterity on
a scale from 1 (lowest) to 9 (highest) was recorded just before the start of and just after the completion
of the 6-week program. The data are shown in the table below.
The program's developers will conduct a hypothesis test for the population mean of paired differences
to see if the program does in fact succeed in improving dexterity.
Before After
Person Program Program Difference
A 6.7 7.6
B 5.4 5.9
C 7 7.6
D 6.6 6.6
E 6.9 7.6
F 7.2 7.6
G 5.5 5.9
H 7.1 7.1
I 7.9 8.2
J 5.9 6.4
K 8.4 8.8
L 6.5 6.6
(A) Have Excel compute the values for the Difference column in the table. Choose the order of subtraction
so that an increase in dexterity has a positive value and a decrease has a negative value.
(B) Write the hypotheses for this test. Be sure to write a sentence explaining what your parameter represents.
H0
Ha
Explanation:
(C) State the conditions needed for the test you are conducting, and verify that the conditions are satisfied.
Insert extra lines as needed.
Click here for a video showing how to insert blank lines.
Click here for a video showing how to check conditions for these kinds of tests.
Condition 1:
Condition 2:
(D) Use Excel formulas to compute the following.
Click here for a video showing how to find the test statistic and p-value for these kinds of tests.
sample est d-bar
null value μ0
sample st dev s
sample size n
null std error
t
df
p-value
(E) Is the sample result statistically significant at the 10% level? Write a sentence explaining what numerical
comparison leads to your answer.
Significant (yes/no)?
Explanation:
(F) Write a conclusion in the context of the problem.
m that people who
mple of 12 people
son’s dexterity on
st after the completion
of paired differences
e order of subtraction
our parameter represents.
onditions are satisfied.
kinds of tests.
plaining what numerical
Math 244 Excel Assignment 5 (EA5)
Hypothesis Testing for Means
Page 3
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Practice questions are available on the last tab of this workbook.
Quest 3 A statistic student at a particular community college noticed that cars in the faculty parking lot tended to
look newer than cars driven by students. She decided to collect sample data on the ages of the cars driven by
faculty members and students, and use a test at the 5% significance level to see if her observation was correct.
She randomly selected 152 faculty cars and recorded how old the cars were. The sample showed a mean
ageof 6.99 years and a standard deviation of 3.65 years. She also randomly selected 217 student cars and
found that these cars had a mean age of 7.49 years and a standard deviation of 3.67 years. Neither sample
had any outliers.
Answer the following questions in order to carry out her hypothesis test.
(A) Identify which group you are calling Population 1 and which you are calling Population 2.
Population 1:
Population 2:
(B) Write the hypotheses for this test. Be sure to write briefly explain what each parameter represents.
H0
Ha
Explanations:
(C) State the conditions needed for the test you are conducting, and verify that the conditions are satisfied.
Insert extra lines as needed.
Click here for a video showing how to insert blank lines.
Click here for a video showing how to check conditions for these kinds of tests.
Condition 1:
Condition 2:
(D) Use the information given in the problem to fill out the following table.
2
n x̄ s s
Sample 1
Sample 2
(E) Pooling IS appropriate here. Explain why.
(F) Use Excel formulas to compute the following.
Click here for a video showing how to use a pooled procedure to find the test statistic and p-value.
sample value x̄1 - x̄2
null value for μ1 - μ2
sp
pooled SE of x̄1 - x̄2
t
df
p-value
(G) Is the sample result statistically significant at the 5% level? Write a sentence explaining what numerical
comparison leads to your answer.
Significant (yes/no)?
Explanation:
(H) Write a conclusion in the context of the problem.
(I) If pooling had not been used for this problem, you would have used different calculations to find the test
statistic and p-value. Perform the unpooled calculations as indicated below. Use the conservative formula
for degrees of freedom.
Click here for a video showing how to use an unpooled procedure to find the test statistic and p-value.
sample value x̄1 - x̄2
null value for μ1 - μ2
unpooled SE of x̄1 - x̄2
tdf
conservative df
p-value
ty parking lot tended to
the ages of the cars driven by
her observation was correct.
sample showed a mean
ted 217 student cars and
.67 years. Neither sample
ameter represents.
onditions are satisfied.
tistic and p-value.
aining what numerical
ulations to find the test
the conservative formula
statistic and p-value.
Math 244 Excel Assignment 5 (EA5)
Hypothesis Testing for Means
Practice
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Example 1 The Carolina Tobacco Company (CTC) advertised that the mean amount if nicotine in its best-selling
cigarette brand is 40 mg. However, Consumer Advocate magazine ran tests of 10 randomly selected
cigarettes for this brand and found the amounts shown in the accompanying list. It's a serious matter
to charge that the company advertising is wrong, so the magazine editor chooses a significance level
of alpha = 0.01 in testing her belief that the mean nicotine content is greater than 40 mg.
mg of nicotine
47.3 43.3
39.3 42.3
40.3 49.3
38.3 40.3
46.3 46.3
(A) Write the hypotheses for this test.
Your answers
H0
Ha
(B) State the conditions needed for the test you are conducting, and verify that the conditions are satisfied.
Click here for a video showing how to check conditions for these kinds of tests.
Condition 1:
Condition 2:
More
below
↓
(C) Use Excel formulas to compute the following.
Click here for a video showing how to find the test statistic and p-value for these kinds of tests.
sample estimate x̄
null value μ0
sample st dev s
sample size n
null std error
t
deg of freedom df
p-value
(D) Write a sentence or two explaining what your p-value means in the context of the problem.
(E) Is the sample result statistically significant at the 5% level? Write a sentence explaining what numerical
comparison leads to your answer.
Significant (yes/no)?
Explanation:
(F) Write a conclusion in the context of the problem.
Example 2 To test whether a fuel additive improves gas mileage for a particular kind of car, the gas mileage
of nine randomly chosen cars was measured (miles per gallon) with and without the additive.
The results shown below.
At alpha = 0.05, can you conclude that the fuel additive improved gas mileage?
No With
Car Additive Additive Difference
1 34.5 36.4
2 36.7 38.8
3 34.4 36.1
4 39.8 40.1
5 33.6 34.7
6 35.4 38.3
7 38.4 40.2
8 35.3 37.2
9 37.9 38.7
(A) Have Excel compute the values for the Difference column in the table. Choose the order of subtraction
so that an improvement in mileage has a positive value and a decrease has a negative value.
(B) Write the hypotheses for this test. Be sure to write a sentence explaining what your parameter represents.
H0
Ha
Explanation:
(C) State the conditions needed for the test you are conducting, and verify that the conditions are satisfied.
Click here for a video showing how to check conditions for these kinds of tests.
Condition 1:
Condition 2:
More
below
↓
(D) Use Excel formulas to compute the following.
Click here for a video showing how to find the test statistic and p-value for these kinds of tests.
sample est d-bar
null value μ0
sample st dev s
sample size n
null std error
t
df
p-value
(E) Is the sample result statistically significant at alpha = 0.05? Write a sentence explaining what numerical
comparison leads to your answer.
Significant (yes/no)?
Explanation:
(F) Write a conclusion in the context of the problem.
Example 3 An advertising executive claims that there is a difference in the mean household income for credit card
holders of Visa Gold and of MasterCard Gold. The results of a random survey of 100 customers from each
group areshown below. The two samples were randomly selected from all Visa Gold and MasterCard Gold
customers, and they were selected independently of each other. There were outliers in each sample,
but no extreme outliers.
Test the executive's claim at the alpha = 0.10 level.
Visa Gold MC Gold
x -bar 60,900 64,300 Values are in dollars per year.
s 12,000 15,000
n 100 100
(A) Identify which group you are calling Population 1 and which you are calling Population 2.
Population 1:
Population 2:
(B) Write the hypotheses for this test. Be sure to write briefly explain what each parameter represents.
H0
Ha
Explanations:
(C) State the conditions needed for the test you are conducting, and verify that the conditions are satisfied.
Click here for a video showing how to check conditions for these kinds of tests.
Condition 1:
Condition 2:
(D) Use the information given in the problem to fill out the following table.
n x̄ s s2
Sample 1
Sample 2
(E) Pooling is NOT appropriate here. Explain why.
(F) Use Excel formulas to compute the following. Use the conservative formula for degrees of freedom.
Click here for a video showing how to use an unpooled procedure to find the test statistic and p-value.
sample value x̄1 - x̄2
null value for μ1 - μ2
unpooled SE of x̄1 - x̄2
t
conservative df
p-value
(G) Is the sample result statistically significant at the 5% level? Write a sentence explaining what numerical
comparison leads to your answer.
Significant (yes/no)?
Explanation:
(H) Write a conclusion in the context of the problem.
(I) If pooling had been used for this problem, you would have used different calculations to find the test
statistic and p-value. Perform the pooled calculations as indicated below.
Click here for a video showing how to use a pooled procedure to find the test statistic and p-value.
sample value x̄1 - x̄2
null value for μ1 - μ2
sp
pooled SE of x̄1 - x̄2
t
df
p-value
cotine in its best-selling
of 10 randomly selected
list. It's a serious matter
oses a significance level
than 40 mg.
Correct answers
H0 μ = 40
Ha μ > 40
the conditions are satisfied.
ts.
Condition 1: The individuals must be selected at random
The question states that this is a random sa
Condition 2: For sample sizes of 30 or more, the only req
For smaller sample sizes, there must be no e
The sample size is 10. The work below sho
does not show strong skewness. This cond
Outliers
Min 0
Q1 1
Med 2
Q3 3
Max 4
IQR
Lower fence
Upper fence
No outliers. The min and max are
within the fences.
hese kinds of tests.
sample estimate x̄ 43.3
null value μ0 40
sample st dev s 3.800585
sample size n 10
null std error 1.20185
t 2.745766
deg of freedom df 9
p-value 0.011317
of the problem.
If it is true that the mean nicotine level for this brand of
that we would see our sample mean of 43.3 mg or some
explaining what numerical
Significant (yes/no)? No
Explanation: The p-value = 0.011, which is g
We fail to reject the null hypothesis in favor of the alter
of 43.3 mg of nicotine per cigarette, this is not far enoug
level. The mean nicotine level for this brand of CTC ciga
the data do not provide strong enough evidence for the
car, the gas mileage
hout the additive.
With
Car No Additive Additive
1 34.5 36.4
2 36.7 38.8
3 34.4 36.1
4 39.8 40.1
5 33.6 34.7
6 35.4 38.3
7 38.4 40.2
8 35.3 37.2
9 37.9 38.7
se the order of subtraction
negative value.
hat your parameter represents.
H0 μd = 0
Ha μd > 0
Explanation: Mu_d is the mean improvement in gas mile
The subtraction is (mileage with additive) - (
the conditions are satisfied.
ts.
Condition 1: The individuals must be selected at random
The question states that this is a random sa
Condition 2: Examine the sample differences.
For sample sizes of 30 or more, the only req
For smaller sample sizes, there must be no e
The sample size is 9. The work below show
strong skewness. The condition is satisfied
Outliers
Min 0
Q1 1
Med 2
Q3 3
Max 4
IQR
Lower fence
Upper fence
No outliers. The min and max are
within the fences.
hese kinds of tests.
sample est d-bar 1.611111
null value μ0 0
sample st dev s 0.770462
sample size n 9
null std error 0.256821
t 6.273295
df 8
p-value 0.00012
explaining what numerical
Significant (yes/no)? Yes
Explanation: The p-value = 0.00012, which i
We reject the null hypothesis in favor of the alternative.
improvement of 1.61 mpg. Under the null hypothesis, w
are significant at alpha = 0.05, so we have convincing ev
the additive does in fact improve mean gas mileage.
hold income for credit card
of 100 customers from each
isa Gold and MasterCard Gold
e outliers in each sample,
in dollars per year.
opulation 2.
Population 1: All Visa Gold card holders
Population 2: All MasterCard Gold card holders
h parameter represents.
H0 μ1 - μ 2 = 0
Ha μ1 - μ 2 ≠ 0
Explanations: Mu_1 is the mean income for all Visa Gold c
Mu_2 is the mean income for all MC Gold ca
the conditions are satisfied.
ts.
Condition 1: The sample should be a independent random
are grouped according to a categorical varia
in a well-designed experiment.
The problem states that each sample was t
Condition 2: For each sample, the following must be che
For sample sizes of 30 or more, the only req
For smaller sample sizes, there must be no e
Each sample size is 100, which is more than
in the samples. The question states that w
n x̄
Sample 1 100 60,900
Sample 2 100 64,300
Pooling is not appropriate because the variances in the t
variance is over 50% larger than the smaller variance.
for degrees of freedom.
test statistic and p-value.
sample value x̄1 - x̄2 -3,400
null value for μ1 - μ2 0
unpooled SE of x̄1 - x̄2 1920.937
t -1.76997
conservative df 99
p-value 0.079811
explaining what numerical
Significant (yes/no)? No
Explanation: The p-value = 0.08, which is m
We fail to reject the null hypothesis in favor of the altern
difference of $3.400 in income, this is not far enough fro
statistically significant at the 5% level. There is not enou
that Visa Gold card holders and MasterCard Gold card h
culations to find the test
t statistic and p-value.
sample value x̄1 - x̄2 -3400
null value for μ1 - μ2 0
sp 13583.08
pooled SE of x̄1 - x̄2 1920.937
t -1.76997
df 198
p-value 0.078271
als must be selected at random from the population.
n states that this is a random sample of cigarettes from CTC's best-selling brand.
zes of 30 or more, the only requirement is that there are no extreme outliers.
ample sizes, there must be no extreme skewness and no outliers.
size is 10. The work below shows there are no outlier. The sample
w strong skewness. This condition is satisfied.
Bins Bins Frequency
38.3 40 40 2 Histogram
40.3 42 42 2 3.5
42.8 44 44 2 3
46.3 46 46 0 2.5
Frequency
49.3 48 48 3 2
1.5
Frequency
2
50 50 1 1.5
6 More 0 1
31.3 0.5
55.3 0
40 42 44 46
The min and max are Bins
sample mean
population mean assuming H0 is true
sample standard deviation
number of individuals in the sample
standard error of x-bar for sample size n
test statistic
degrees of freedom df = n - 1
significance level of the observed results assuming H0 is true
nicotine level for this brand of cigarettes is 40 mg, there is a 0.011 probability
mple mean of 43.3 mg or something more extreme.
The p-value = 0.011, which is greater than alpha = 0.01.
hypothesis in favor of the alternative. Although our sample had a mean
r cigarette, this is not far enough above 40 mg to be significant at the 1%
level for this brand of CTC cigarette may in face be more than 40, but
trong enough evidence for the magazine editor to make this claim in print.
Difference
1.9
2.1
1.7
0.3
1.1
2.9
1.8
1.9
0.8
mean improvement in gas mileage for all cars when the additive is used.
on is (mileage with additive) - (mileage without additive).
als must be selected at random from the population.
n states that this is a random sample of cars.
sample differences.
zes of 30 or more, the only requirement is that there are no extreme outliers.
ample sizes, there must be no extreme skewness and no outliers.
size is 9. The work below shows there are no outliers. The histogram shows no
ness. The condition is satisfied.
Bins Bins Frequency
0.3 0 0 0 Histogram
1.1 0.4 0.4 1 5
1.8 0.8 0.8 0 4
Frequency
1.9 1.2 1.2 2 3
2.9 1.6 1.6 0 2
2 2 4
1
0.8 2.4 2.4 1
0
-0.1 2.8 2.8 0
3.1 3.2 3.2 1
More 0 Bins
The min and max are
mean of the sample differences
population mean difference assuming H0 is true
sample standard deviation
number of individuals in the sample
standard error of x-bar for sample size n
test statistic
degrees of freedom df = n - 1
significance level of the observed results assuming H0 is true
The p-value = 0.00012, which is less than alpha = 0.05.
hesis in favor of the alternative. In our sample, cars using the additive had a mean
g. Under the null hypothesis, we expected no improvement. Our sample results
0.05, so we have convincing evidence that the null hypothesis is wrong and that
mprove mean gas mileage.
card holders
rd Gold card holders
mean income for all Visa Gold card holders
mean income for all MC Gold card holders
hould be a independent random samples from two populations, one random sample where
according to a categorical variable, or individuals are randomly assigned to treatment groups
gned experiment.
states that each sample was taken randomly and independently of each other.
mple, the following must be checked:
zes of 30 or more, the only requirement is that there are no extreme outliers.
ample sizes, there must be no extreme skewness and no outliers.
size is 100, which is more than 30. We need to know there are no extreme outliers
es. The question states that while there were outliers, none were extreme.
s s2
12,000 144,000,000
15,000 225,000,000
e because the variances in the table above (s^2) are so different. The larger
er than the smaller variance.
difference of sample means
mean value for x̄1 - x̄2 assuming H0 is true
standard error of x̄1 - x̄2 assuming unequal population variances
test statistic
degrees of freedom using the conservative formula min(n1, n2) - 1
significance level of the observed results assuming H0 is true
The p-value = 0.08, which is more than alpha = 0.05.
hypothesis in favor of the alternative. Although our samples showed a mean
come, this is not far enough from the null value of $0 (no difference) to be
the 5% level. There is not enough evidence to demonstrate conclusively
rs and MasterCard Gold card holders have different mean incomes.
difference of sample means
mean value for x̄1 - x̄2 assuming H0 is true
pooled standard deviation
standard error of x̄1 - x̄2 assuming equal population variances
test statistic
degrees of freedom
significance level of the observed results assuming H0 is true
48 50