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Math 102: Elements of Statistics
Answer Key for ”Mile Shoes” Worksheet
Part 1
The FastTrack shoe company manufactures a shoe called the Scorcher and wants to
advertise that it makes runners faster. They select 50 random high school runners
and give them Scorchers to wear. The company has the students run a timed mile
in their old shoes, rest, and then run a second timed mile in the Scorchers. The
company wants to claim that the shoes have lowered the students’ running times.
1. Describe the Population Characteristic
Let µd be the average population difference of running times in people’s
regular shoes vs running in a pair of Scorchers. We want to know if µd is
greater than 0.
Notice this is an example of a paired data sample because each individual
runner is providing a before and after running time.
2. Set up H0
H0 : we assume that µd = 0
3. Set up Ha
Ha : µd > 0 (one-tailed test)
4. Select Significance Level
We select α = 0.05 as the significance level. This is a good compromise
between requiring sufficient evidence but not being overly restrictive.
5. Display test statistic
We compute a t value based on:
t=
x¯d − µd
,
σx¯d
sd
σx¯d = √
n
The t value gives us a measure of how far away from the assumed population
mean our actual sample falls.
6. Check Assumptions
• We want to be sure that n is large or that the population distribution is
normal. It is reasonable to conclude that mile times are close to normal
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distribution. We can verify this with a normality plot of the sample
data. We also choose a sample size of n = 50 which is greater than 30.
So the sampling distribution will be normal anyway.
• We want our sample to be random. We assume we have a process to
randomly select runners for our sample.
• We want our sample size to be less than 10% of the population. Since
n = 50 is less than 10% of the population (millions of runners?) we
are ok.
7. Compute values for test statistic:
From the data provided in the spreadsheet:
x¯d = 0.26, sd = 0.103
sd
σx¯d = √ = 0.015
n
x
¯−µ
0.26 − 0.00
t=
=
= 17.795
σx¯
0.015
8. Determine P-value
We consult Table 4 in our book. Since we are testing for ”>” here we need
to use a one-tailed test. We compute the P-value as 0.00. Note we can also
use the formula TDIST() in the spreadsheet instead of Table 4.
9. Conclusion
We compare our P-value of 0.000 against our significance level of α = 0.05.
The P-value is less, so we reject the null hypothesis.
Conclusion: reject H0 .
There is sufficient evidence that the FastTrack corporation concludes the
Scorchers make runners faster. They contact their advertising company with
the results and give them the go-ahead for the full page ad in Runner’s World
magazine.
Part 2
An independent exercise lab is not convince by FastTrack’s claim. They know that
most people will run a second mile faster than their first. So the observed difference
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in the data is explained by this phenomena, not by the Scorchers. They decide to
replicated the experiment to determine if people are significantly faster in a second
mile, regardless of shoes.
1. Describe the Population Characteristic
Let µd be the average population difference of running times of people in
their first to second mile (a positive average means faster time in the second
mile). We want to know if µd is greater than 0.
Notice this is an example of a paired data sample because each individual
runner is providing a before and after running time.
2. Set up H0
H0 : we assume that µd = 0
3. Set up Ha
Ha : µd > 0 (one-tailed test)
4. Select Significance Level
We select α = 0.05 as the significance level. This is a good compromise
between requiring sufficient evidence but not being overly restrictive.
5. Display test statistic
We compute a t value based on:
t=
x¯d − µd
,
σx¯d
sd
σx¯d = √
n
The t value gives us a measure of how far away from the assumed population
mean our actual sample falls.
6. Check Assumptions
• We want to be sure that n is large or that the population distribution is
normal. It is reasonable to conclude that mile times are close to normal
distribution. We can verify this with a normality plot of the sample
data. We also choose a sample size of n = 50 which is greater than 30.
So the sampling distribution will be normal anyway.
• We want our sample to be random. We assume we have a process to
randomly select runners for our sample.
• We want our sample size to be less than 10% of the population. Since
n = 50 is less than 10% of the population (millions of runners?) we
are ok.
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7. Compute values for test statistic:
From the data provided in the spreadsheet:
x¯d = 0.22, sd = 0.127
sd
σx¯d = √ = 0.018
n
0.22 − 0.00
x
¯−µ
=
t=
= 12.403
σx¯
0.018
8. Determine P-value
We consult Table 4 in our book. Since we are testing for ”>” here we need
to use a one-tailed test. We compute the P-value as 0.00. Note we can also
use the formula TDIST() in the spreadsheet instead of Table 4.
9. Conclusion
We compare our P-value of 0.000 against our significance level of α = 0.05.
The P-value is less, so we reject the null hypothesis.
Conclusion: reject H0 .
There is sufficient evidence to reject H0 and to conclude that people are
faster in the second mile. The independent corporation threatens to sue FastTrack unless they rescind their ad.
Part 3
FastTrack is not done yet. They randomly select 100 students and have 50 run a
mile in their regular shoes and 50 others run a mile in Scorchers. They want to
show that the Scorcher group is faster.
1. Describe the Population Characteristic
Let µR be the average mile time for people wearing their ”Regular” shoes
and let µS be the average mile time for people wearing the ”Scorchers”.
FastTrack wants to show that people are faster wearing Scorchers.
This is an example of independent samples because the 50 runners in the
control group (old shoes) are not at all connected to the 50 runners in the test
group (Scorchers).
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2. Set up H0
H0 : we assume that the two populations are the same:
H0 : µR − µS = 0
3. Set up Ha
Ha : µR − µS > 0 (one-tailed test)
4. Select Significance Level
We select α = 0.05 as the significance level. This is a good compromise
between requiring sufficient evidence but not being overly restrictive.
5. Display test statistic
We compute a t value based on:
s2R
nR
s2
VS = S
nS
(VR + VS )2
VR =
df =
VR2
NR −1
+
VS2
NS −1
p
σ = VR + VS
¯R − x
X
¯S − 0
t=
σ
The t value gives us a measure of how far away from the assumed population
mean our actual sample falls.
6. Check Assumptions
• We want to be sure that n is large or that the population distribution is
normal. It is reasonable to conclude that mile times are close to normal
distribution. We can verify this with a normality plot of the sample
data. We also choose a sample size of n = 50 which is greater than 30.
So the sampling distribution will be normal anyway.
• We want our sample to be random. We assume we have a process to
randomly select runners for our sample.
• We want our sample size to be less than 10% of the population. Since
n = 50 is less than 10% of the population (millions of runners?) we
are ok.
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7. Compute values for test statistic:
From the data provided in the spreadsheet:
x
¯R = 4.98, sR = 0.520
x
¯S = 4.92, sS = 0.509
s2
VR = R = 5.415 × 10−3
nR
s2
VS = S = 5.179 × 10−3
nS
(VR + VS )2
df = V 2
= 97
VS2
R
+
NR −1
NS −1
p
σ = VR + VS = 0.103
¯R − x
X
¯S − 0
t=
= 0.552
σ
8. Determine P-value
We consult Table 4 in our book. Since we are testing for ”>” here we need
to use a one-tailed test. We compute the P-value as 0.291. Note we can also
use the formula TDIST() in the spreadsheet instead of Table 4.
9. Conclusion
We compare our P-value of 0.291 against our significance level of α = 0.05.
The P-value is greater, so we fail to reject the null hypothesis.
Conclusion: fail to reject H0 .
There is not sufficient evidence to conclude that the Scorcher group is faster.
The FastTrack corporation sees there is a slight performance improvement
and is thus motivated to select a larger sample so they can prove their assertion. What do you think about their plan?
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