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EX P LO RAT IO N 6.1A : Haircut Prices
Haircut Prices
EXPLORATION
Do women pay more than men for haircuts? Is this a statistical tendency or always true? By
how much do women spend more than men, on average? How much do haircut prices vary
within a sex as well as between sexes?
To investigate these questions a professor asked students in her class to report the cost of
their most recent haircut, along with their sex.
1. Which would you consider to be the explanatory variable and which the response? Also
classify the type (categorical or quantitative) for each variable.
Explanatory:
Type:
Response:
Type:
2. Is this an experiment or an observational study? Explain briefly.
3. Did the professor who collected the data make use of random sampling, random assign-
ment, both, or neither?
Sex
The following “parallel” dotplots (using the same scale along the horizontal axis) reveal
the sample distributions of haircut prices for each sex:
Female
Male
0
25
50
75
100
Haircut cost (in dollars)
125
150
4. Compare and contrast the distributions of haircut prices between men and women in this
class. (Hint: As you learned in the Preliminaries, comment on center, variability, shape,
and unusual observations. You should give enough detail that someone reading your comments could re-create the overall pattern of the graphs from your description. Also be sure
to relate your comments to the context.)
5. Further explore the data.
a. Explain why the right skewness of these distributions makes sense in this context.
6.1A
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CH A PT E R 6
Comparing Two Means
b. Explain why the unusual behavior at the lower end of the dotplots—several values at $0
and then a gap to the next smallest prices—makes sense in this context.
6. Based on the dotplots (without performing any calculations), make a guess for each sex’s
mean haircut price.
Men:
Women:
7. Based on the dotplots (without performing any calculations), which sex do you think has
the larger standard deviation of haircut prices? Explain your answer.
8. Copy the haircut data, which you can access from the book’s website, to the clipboard.
Open the Descriptive Statistics applet. Check the Stacked check box (notice the applet
assumes the explanatory variable is the first column and the response variable is the
second column.) Also keep the Includes header box checked and press Clear. Paste
the data into the Sample data box and press Use Data. You should see that the dotplots
are similar to the ones shown previously. Check the Actual boxes to show the means and
standard deviations (Std dev).
a. Report the sample size, mean haircut price, and standard deviation (SD) of haircut
prices for each sex. (Include appropriate symbols and measurement units.)
Sample size
Sample mean
Sample SD
Men
Women
b. Which sex has the larger mean haircut price? Is this what you predicted in #6?
c. Which sex has the larger SD of haircut prices? Is this what you predicted in #7?
9. Would you conclude that these data show an association between haircut price and a
person’s sex? If so, describe the nature of this association.
To enter special characters and
formatting, use “Ctrl+E” in
Windows and “Cmd + E” in
Mac.
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EX P LO RAT IO N 6.1A : Haircut Prices
10. Let’s return to the research questions that we started with. Address these questions based
on the previous graphs and statistics.
a. Do women pay more than men for haircuts? If so, is this a statistical tendency (i.e., true
on average) or always true?
b. By how much do women spend more than men, on average?
11. Do the unequal sample sizes between the two sexes lead you to doubt whether any con-
clusions can be drawn from these data? Explain.
12. Based on how these data were collected, would you feel comfortable generalizing your
results to the population of all college students in the U.S.? How about the population of
all college students at the professor’s university? Explain your answers.
13. Based on both the differences in centers and the amount of overlap in the distributions
of haircut prices between men and women, do you predict that the difference will turn
out to be statistically significant? (You will learn how to assess statistical significance in
the next section.)
Further analyses
The individual values of the haircut prices that you have been analyzing are:
Women (n =
37):
0, 0, 0, 15, 15, 15, 20, 20, 20, 25, 30, 30, 35, 35, 35, 40, 45, 45, 45,
45, 50, 50, 50, 50, 55, 60, 65, 70, 70, 75, 90, 110, 120, 120, 150,
150, 150
Men (n = 13):
0, 0, 0, 14, 15, 15, 20, 20, 20, 22, 23, 60, 75
As you learned in Section 3.2, the median is the middle value in a data set once the values are
arranged in order. The location of the median can be found by calculating (n + 1)2.
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CH A PT E R 6
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Comparing Two Means
14. Now let’s look at the median haircut prices and compare them to the mean haircut price.
a. Determine (by hand) the median haircut price for each sex. (You can verify your calcu-
lation by checking the Actual box for the Median in the applet.)
Median haircut price for women:
Median haircut price for men:
b. Which sex has the larger median haircut price? Is this what you expected?
Definitions
c. Are the medians less than or greater than the means? Is this consistent with the right-
skewed distributions of haircut prices? Explain.
The value for which 25%
of the data lie below
that value is called the
lower quartile (or 25th
percentile). Similarly, the
value for which 25% of the
data lie above that value is
called the upper quartile
(or 75th percentile). Quartiles can be calculated by
determining the median of
the values above/below
the location of the overall
median. The difference
between the quartiles is
called the inter-quartile
range (IQR), another measure of variability along
with standard deviation.
The five-number
summary for the distribution of a quantitative
variable consists of the
minimum, lower quartile,
median, upper quartile,
and maximum.
One way to summarize the distribution of a quantitative variable such as haircut price is by
dividing the distribution into four pieces of roughly equal size (number of observations). In
other words, summarize the distribution by determining where the bottom 25% of the data
are, the next 25%, the next 25%, and then the top 25%.
15. Explore the data using quartiles and the five-number summary.
a. Calculate the lower quartile for the women’s haircut prices. First note that the median is
the (37 + 1)2 = 19th ordered value. So, the lower quartile is the median of the bottom
18 values, which is found in position (18 + 1)2 = 9.50. So, the lower quartile is the
average of the 9th and 10th ordered values from the bottom.
b. Similarly, calculate the upper quartile for the women’s haircut prices.
c. Calculate the lower and upper quartiles for the men’s haircut prices.
16. Report the five-number summary for the women’s haircut prices and for the men’s hair-
cut prices:
Minimum
Women’s haircut prices
Men’s haircut prices
Lower
quartile
Median
Upper
quartile
Maximum
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EX P LO RAT IO N 6.1A : Haircut Prices
17. In the applet, check the box for Boxplot to overlay the two boxplots as well. Describe what
the boxplots reveal about how the distributions of haircut prices compare between male
and female students in the professor’s class.
18. Find the IQR for both the men’s and women’s haircut prices.
One advantage to the IQR is that it is not sensitive to extreme values/outliers like the standard deviation is. Just like the median is not sensitive to extreme values/outliers, but the
mean is.
KEY IDEA
The IQR is a resistant measure of variability, whereas the standard deviation is sensitive to extreme values and skewness.
19. Remove the male haircut price of $75 from the data and (in the applet) find the new SD
and IQR. Compare them to the values you had earlier and confirm that this illustrates that
the IQR is more resistant to extreme values than the SD.
More data
20. Collect data from yourself and your classmates on most recent haircut price and sex.
Enter the data into the applet and examine dotplots, boxplots, and summary statistics.
Write a paragraph or two summarizing what the data reveal about whether haircut price
is associated with the sex of your classmates. Also address the research questions with
which this exploration began. Finally, comment on how broadly you would be willing to
generalize your findings.
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Definition
A boxplot is a visual
display of the five-number summary. The box
displays the middle 50%
of the distribution and
its width (the IQR) helps
us compare the spread
of the distribution; the
whiskers extend to the
smallest and largest values in the data set.
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EX P LO RAT IO N 6.2: Lingering Effects of Sleep Deprivation
Lingering Effects of Sleep Deprivation
STEP 1: Ask a research question. Many students pull “all-nighters” when they have
an important exam or a pressing assignment. Concerns that may arise include: Can you really
function well the next day after a sleepless night? What about several days later: Can you recover
from a sleepless night by getting a full night’s sleep on the following nights?
1. What is your research conjecture about whether or not one can recover from a sleepless
night by getting a full night’s sleep on the following nights? What are some other related
questions that you would be interested in investigating related to this issue?
STEP 2: Design a study and collect data. Researchers Stickgold, James, and Hobson
investigated delayed effects of sleep deprivation on learning in a study published in Nature
Neuroscience (2000). Twenty-one volunteers, aged 18–25 years, were first trained on a visual
discrimination task that involved watching stimuli appear on a computer screen and reporting
what was seen. See Figure 6.9.
Subjects were flashed the screen on the left and then it was masked by
the screen on the right. Then subjects were asked whether they had seen an L or a V and
whether the slanted lines were placed vertically or horizontally.
FIGURE 6.9
After the training period, subjects were tested. Performance was recorded as the minimum time (in milliseconds) between the appearance of stimuli and an accurate response. Following these baseline measurements, one group was randomly assigned to be deprived of sleep
for 30 hours, followed by two full nights of unrestricted sleep, whereas the other group was allowed to get unrestricted sleep on all three nights. Following this, both groups were retested on
the task to see how well they remembered the training from the first day. Researchers recorded
the improvement in performance as the decrease in time required at retest compared to training.
(Note: For example, if someone took 5 milliseconds (ms) to respond at the beginning of the
study and then 2 ms to respond at the end, the improvement score is 3 ms. But if someone took
2 ms at the beginning and then 5 ms at the end, the improvement score is −3 ms.)
The goal of the study was to see whether the improvement scores tend to be higher for the
unrestricted sleep treatment than for the sleep deprivation treatment.
EXPLORATION
6.2
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CH A PT E R 6
Comparing Two Means
2. Identify the explanatory and response variables in this study. Also classify them as either
categorical or quantitative.
Explanatory:
Type:
Response:
Type:
3. Was this an experiment or an observational study? Explain how you are deciding.
4. Let 𝜇unrestricted be the long-run mean improvement on this task three days later when
someone has had unrestricted sleep and let 𝜇deprived denote the long-run mean improvement when someone is sleep deprived on the first night.
In words and symbols, state the null and the alternative hypotheses to investigate whether
sleep deprivation has a negative effect on improvement in performance on visual discrimination tasks. (Hint: For the alternative hypothesis: Do you expect the people to do better
or worse when sleep deprived? Based on your answer, what sign/direction should you
choose for the alternative hypothesis?)
Here are the data, with positive values indicating better performance at retest than at training,
and negative values indicating worse performance at retest than at training:
Unrestricted-sleep group’s improvement scores (milliseconds):
25.20, 14.50, −7.00, 12.60, 34.50, 45.60, 11.60, 18.60, 12.10, 30.50
Sleep-deprived group’s improvement scores (milliseconds):
−10.70, 4.50, 2.20, 21.30, −14.70, −10.70, 9.60, 2.40, 21.80, 7.20,
10.00
STEP 3: Explore the data.
5. To look at graphical and numerical summaries of the data from the study, go to the
Multiple Means applet. The sleep deprivation data have already been entered into the applet.
a. Notice that the applet creates parallel dotplots, one for each study group. Based on
these dotplots alone, which group (unrestricted or deprived) appears to have had the
higher mean improvement? How are you deciding?
b. Based on the dotplots alone, which group (unrestricted or deprived) appears to have
had more variability in improvement? How are you deciding?
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EX P LO RAT IO N 6.2: Lingering Effects of Sleep Deprivation
c. Notice also that the applet also computes numerical summaries of the data, such as the
mean and standard deviation (SD) for the improvements in each group.
i. For the unrestricted group, record the sample size (n), mean, and SD.
n u n re s t r i c t e d
=
xunrestricted =
SD unrestricted
=
ii. For the deprived group, record the sample size (n), mean, and SD.
ndeprived =
xdeprived =
SDdeprived =
Recall from earlier in the course that the standard deviation is a measure of variability.
Relatively speaking, smaller standard deviation values indicate less variability and a distribution whose data values tend to cluster more closely together, compared to a distribution with
a larger standard deviation.
d. Based on the numerical summaries reported in #5(c), which group (unrestricted or
deprived) had the higher mean improvement?
e. Based on the numerical summaries reported in #5(c), which group (unrestricted or
deprived) had the higher variability in improvement?
f. Notice that the applet also reports the observed difference in means for the improve-
ments of the two groups. Record this value (and its measurement units).
xunrestricted − xdeprived =
g. Before you conduct an inferential analysis, does this difference in sample means (as
reported in #5(f)) strike you as a meaningful difference? Explain your answer.
STEP 4: Draw inferences.
6. What are two possible explanations for why we observed the two groups to have different
sample means for improvement in performance?
7. Describe how you might go about deciding whether the observed difference between the
two sample means is statistically significant. (Hint: Think about how you assessed whether
an observed difference between two sample proportions was statistically significant in
Chapter 5. Use the same strategy, with an appropriate modification for working with
means instead of proportions.)
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CH A PT E R 6
Comparing Two Means
Once again the key question is how often random assignment alone would produce a difference
in the groups at least as extreme as the difference observed in this study if there really were no
effect of sleep condition on improvement score. You addressed similar questions in Chapter 5
when you analyzed the dolphin therapy and yawing studies. The only change is that now the
response variable is quantitative rather than categorical, so the relevant statistic is the difference
in group means rather than the difference in group proportions. Also once again, we use simulation to investigate how often such an extreme difference would occur by chance (random
assignment) alone (if the null hypothesis of no difference/no effect/no association were true).
In other words, we will again employ the 3S strategy.
1. Statistic:
8. A natural statistic for measuring how different the observed group means are from each
other is the difference in the mean improvement scores between the two groups. Report
the value of this statistic, as you did in #5(f).
2. Simulate: You will start by using index cards to perform a tactile simulation of randomly assigning the 21 subjects between the two groups, assuming that sleep condition has
no impact on improvement.
Because the null hypothesis asserts that improvement score is not associated with sleep
condition, we will assume that the 21 subjects would have had exactly the same improvement
scores as they did, regardless of which sleep condition group (unrestricted or deprived) the
subject had been assigned.
9. a. How many index cards do you need to conduct this simulation?
b. What will you write on each index card?
To conduct one repetition of this simulation:
• Shuffle the stack of 21 cards well and then randomly distribute cards into two stacks: one
stack with 10 cards (the unrestricted group) and one with 11 (the sleep-deprived group).
• Calculate and report the sample means for each rerandomized group:
Rerandomized unrestricted group’s mean:
Rerandomized deprived group’s mean:
• Calculate the difference in group means: unrestricted mean minus sleep-deprived mean.
Report this value.
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EX P LO RAT IO N 6.2: Lingering Effects of Sleep Deprivation
• Combine this result with your classmates’ to create a dotplot that shows the distribution
of several possible values of difference in sample means that could have happened due to
pure chance if sleep condition has no impact on improvement. Sketch a dotplot on an axis
like the one below, being sure to label the horizontal axis.
−20
−15
−10
−5
0
5
10
15
Label:
c. At about what value is the dotplot centered? Explain why this makes sense. (Hint: What
are we assuming to be true when we conduct the simulation?)
d. Where is the observed difference in means from the original study (as reported in #8)
on the dotplot? Did this value happen often, somewhat rarely, or very rarely? How are
you deciding?
10. As before with simulation-based analyses, you would now like to conduct many, many
more repetitions to determine what is typical and what is not for the difference in group
means, assuming that sleep condition has no impact on improvement score. We think
you would prefer to use a computer applet to do this rather than continue to shuffle cards
for a very long time, calculating the difference of group means by hand. Go back to the
Multiple Means applet, check the Show Shuffle Options box, select the Plot display, and
press Shuffle Responses.
a. Describe what the applet is doing and how this relates to your null hypothesis from #4.
b. Record the simulated difference in sample means for the rerandomized groups, as
given in the applet output. Is this difference more extreme than the observed difference
from the study (as reported in #8)? How are you deciding?
c. Click on Shuffle Responses again and record the simulated difference in sample means
for the rerandomized groups. Did it change from #10(b)?
20
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CH A PT E R 6
Comparing Two Means
d. Click on Re-Randomize again and record the simulated difference in sample means
for the rerandomized groups. Did it change from #10(b) and #10(c)?
e. Now to see many more possible values of the difference in sample means, assuming
sleep condition has no impact on improvement, do the following in the Multiple
Means applet:
• Change Number of Shuffles from 1 to 997.
• Press Shuffle Responses to produce a total of 1000 shuffles and rerandomized
statistics.
f. Consider the histogram of the 1,000 could-have-been values of difference in sample
means, assuming that sleep condition has no effect on improvement.
i. What does one observation on the histogram represent? (Hint: Think about what
you would have to do to put another observation on the graph.)
ii. Describe the overall shape of the null distribution displayed in this histogram.
iii. Where does the observed difference in sample means (as reported in #8) fall in this
histogram: near the middle or out in a tail? Are there a lot of observations that are
even more extreme than the observed difference, assuming sleep condition has no
impact on improvement? How are you deciding?
g. To estimate a p-value, continue with the Multiple Means applet.
• Type in the observed difference in group means (as reported in #8) in the Count
Samples box (for the one-sided alternative hypothesis) and press Count.
• Record the approximate p-value.
h. Fill in the blanks of the following sentence to complete the interpretation of the p-value.
The p-value of _______ is the probability of observing _____________
__________________ assuming ______________________________.
3. Strength of evidence:
11. Based on the p-value, evaluate the strength of evidence provided by the experimental data
against the null hypothesis that sleep condition has no effect on improvement score: not
much evidence, moderate evidence, strong evidence, or very strong evidence?
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EX P LO RAT IO N 6.2: Lingering Effects of Sleep Deprivation
12. Significance: Summarize your conclusion with regard to strength of evidence in the con-
text of this study.
13. Estimation:
a. Use the 2SD method to approximate a 95% confidence interval for the difference in
long-run mean improvement score for subjects who get unrestricted sleep minus the
long-run mean improvement score for subjects who are sleep deprived. (Hints: Remember the observed value of the difference in group means and obtain the SD of the
difference in group means from the applet’s simulation results. The interval should be
observed difference in means ±2SD, where SD represents the standard deviation of
the null distribution of the difference in group means.)
b. Interpret what this confidence interval reveals, paying particular attention to whether
the interval is entirely positive, entirely negative, or contains zero. (Hint: Be sure to
convey “direction” in your interpretation by saying how much larger improvement
scores are on average for the treatment you find to have the larger long-run mean: I’m
95% confident that the long-run mean improvement score is __________ to ________
higher with the _________ treatment as opposed to the _________ treatment.)
STEP 5: Formulate conclusions.
14. Generalization: Were the participants in this study randomly selected from a larger pop-
ulation? Describe the population to which you would feel comfortable generalizing the
results of this study.
15. Causation: Were the participants in the study randomly assigned to a sleep condition?
How does this affect the scope of conclusion that you can draw?
Another statistic
Could we have chosen a statistic other than the difference in group means to summarize how
different the two groups’ improvement scores were? Yes, for example we could have used
the difference in group medians. Why might we do this? For one reason, the median is less
affected by outliers than the mean (see Section 3.2).
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To analyze the difference in group medians, we carry out the 3S strategy as before, except:
• We calculate the observed value of the difference in medians as the statistic.
• After we conduct the rerandomizing, we calculate the difference in medians for the rerandomized data. Then we repeat this process a large number of times.
• We determine the p-value by counting how many of the simulated statistics are at least as
large as the observed value of the difference in medians.
16. Return to the Multiple Means applet and use the Statistic pull-down menu (on the left)
to select Difference in Medians.
a. From the Summary Statistics for the original data, record the median improvement
score for each group. Also record the difference between the medians (unrestricted
median minus deprived median).
Unrestricted median:
Deprived median:
Difference (unrestricted − deprived):
b. Enter 1000 for Number of Shuffles and press Shuffle Responses. Describe the result-
ing null distribution of difference in group medians. Does this null distribution appear
to be centered near zero? Does it seem to have a bell-shaped distribution?
c. To calculate a p-value based on the difference in medians, enter the observed value in
the Count samples box. Then press Count. Report both the value that you enter into
the applet and the resulting p-value.
d. Does this p-value indicate strong evidence that sleep deprivation has a harmful effect
on improvement score? Explain how you are deciding.
e. With which statistic (difference in means or difference in medians) do the data provide
stronger evidence that sleep deprivation has a harmful effect on improvement score?
Explain how you are deciding.
17. STEP 6: Look back and ahead.
Looking back: Did anything about the design and conclusions of this study concern you?
Issues you may want to critique include:
• Any mismatch between the research question and the study design
• How the experimental units were selected
• How the treatments were assigned to the experimental units
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EX P LO RAT IO N 6.2: Lingering Effects of Sleep Deprivation
• How the measurements were recorded
• The number of experimental units in the study
• Whether what we observed is of practical value
Looking ahead: What should the researchers’ next steps be to fix the limitations or build
on this knowledge?
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EX P LO RAT IO N 6.3 : Close Friends
Close Friends
How many close friends do you have? You know, the kind of people you like to talk to about
important personal matters. Do men and women tend to differ on the number of close
friends? And if so, by how much do men and women differ with regard to how many close
friends they have? One of the questions asked of a random sample of adult Americans on the
2004 General Social Survey (GSS) was:
From time to time, most people discuss important matters with other people. Looking back
over the last six months—who are the people with whom you discussed matters important
to you? Just tell me their first names or initials.
The interviewer then recorded how many names each person gave, along with keeping
track of the person’s sex. The GSS is a survey of a representative sample of U.S. adults who are
not institutionalized.
1. Identify the variables recorded. Also classify each as either categorical or quantitative, and
identify each variable’s role: explanatory or response.
2. Did this study make use of random assignment, random sampling, both, or neither?
3. Was this an experiment or an observational study? Explain how you are deciding.
4. In words, state the null and the alternative hypotheses to test whether American men and
women differ with regard to how many friends they have.
5. Define the parameters of interest and assign symbols to them.
6. State the null and the alternative hypotheses in symbols.
EXPLORATION
6.3
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The survey responses are summarized in the Table 6.5 (and the data can be found in expanded
form in the file CloseFriends).
Table 6.5 Number of close friends reported by sex
Number of close friends reported
0
1
2
3
4
5
6
Total
Men
196
135
108
100
42
40
33
654
Women
201
146
155
132
86
56
37
813
7. Now you will obtain numerical summaries (statistics, such as mean and SD) of the survey
data using the Multiple Means applet.
• Open the data file CloseFriends to access the raw data. Copy the data (e.g., CTRL-A
and CTRL-C).
• Open the Multiple Means applet and press Clear. Click inside the Sample data box
and paste (e.g., CTRL-V). Then press Use Data.
a. Report the sample size, sample mean, and sample SD of the number of close friends
reported for each sex. Based on the sample statistics, who tends to have more close
friends, on average: men or women? How are you deciding?
b. Based on the sample statistics, who tends to have more variability with respect to how
many close friends they have: men or women? How are you deciding?
c. Based on the data as presented in Table 6.5, are the distributions of number of close
friends symmetric or skewed? If skewed, in which direction?
d. Calculate the observed difference in the mean number of close friends between these two
groups (men − women). (Verify your calculation by checking the Observed diff output.)
e. Notice that the value recorded in (d) is a negative number. Well, number of friends can
only be 0 or more. Why is the number recorded in (d) less than 0?
The question we want to answer is, Is the difference in mean number of close friends between
men and women, as seen in the GSS sample data, something that could plausibly have happened
by chance, by random sampling alone, if, on average, men and women have the same number
of close friends?
8. Using the same approach as in Section 6.2 with the Multiple Means applet, you can use
this applet to generate possible values of the difference in sample means under the null
hypothesis by shuffling which response values go with which explanatory variable values.
Check the Show Shuffle Options box and enter 1,000 for the Number of Shuffles. Press
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EX P LO RAT IO N 6.3 : Close Friends
the Shuffle Responses button (there will be a pause; this is a large data set!). The histogram on the right is the simulated null distribution for the difference in sample means.
Enter the observed difference in sample means in the Count Samples box and press
Count.
a. Record your estimated p-value.
b. Fill in the blanks of the following sentence, to complete the interpretation of the
p-value.
The p-value of ________ is the probability of observing
________________________________ assuming
__________________________________.
9. If the difference in means was larger (more different than 0), how would this impact the
size of the p-value?
10. How would increasing the sample size (all else remaining the same) change the p-value?
Why?
11. How would increasing the standard deviation of the number of close friends for both
males and females (all else remaining the same) change the p-value? Why?
As we saw earlier, another measure of the strength of evidence against the null hypothesis
would be to standardize the statistic by subtracting the hypothesized value and dividing by
the standard error of the statistic.
12. On the left side of the applet (below the Sample data window) use the Statistic pull-
down menu to select the t-statistic (again, there will be a pause). Record the value of
the t-statistic that is computed for your data. Write a one-sentence interpretation of this
value.
Notice that the standardized statistic uses the letter t (the one you should remember from
Chapter 3) instead of the z (from Chapters 1 and 5) we saw when testing proportions. This
is because the theoretical distribution used is now a t-distribution instead of a normal distribution. These t-distributions are very similar to normal distributions especially when sample
sizes are large. The t-statistic, like the z-statistic, tells us how many standard deviations our
sample difference is above or below the mean and it can be judged in the same manner. More
details about the t-statistic are given earlier in this section and in the Calculation Details
appendix.
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13. In light of the value of the standardized statistic, should you expect the p-value to be large
or small? How are you deciding?
14. The histogram on the far right now displays the null distribution of simulated t-statistics.
Describe the behavior of this distribution.
15. Below the null distribution check the box Overlay t distribution. Does the t-distribution
appear to adequately predict the behavior of the shuffled t-statistics? What do you think
this suggests about whether the validity conditions will be met for these data?
16. Enter the observed value of the t-statistic in the Count Samples box and press Count.
(Remember to use the pull-down menu to specify what direction(s) you want to consider more extreme based on the alternative hypothesis.) How does the p-value from the
t-distribution (in orange) compare to the simulation-based p-value (from #8(a))?
Validity conditions
The validity conditions required for this theory-based approach (a “two-sample t-test”) to be
valid are shown in the box.
VA L I D I T Y C O N D I T I O N S
The quantitative variable should have a symmetric distribution in both groups or you
should have at least 20 observations in each group and the sample distributions
should not be strongly skewed.
17. Do the validity conditions appear to be satisfied for these data? Justify your answer.
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EX P LO RAT IO N 6.3 : Close Friends
18. Based on the above analysis, state your conclusion in the context of the study. Be sure to
comment on the following.
a. Statistical significance: Do the data provide evidence that the mean number of close
friends that men have is different from the mean number of friends women have? How
are you deciding?
b. Causation: Do the data provide evidence that how many friends one has is caused by
one’s sex? How are you deciding?
c. Generalization: To whom can you apply the results of this study? All people? All adults?
How are you deciding?
19. Estimation: When you have selected the t-statistic and the theory-based overlay, you can
also use this applet for estimating the parameter of interest. (You can also go directly to
the Theory-Based Inference applet which will also allow you to change the confidence
level.) Check the box next to 95% CI(s) for difference in means.
a. Identify, in words related to the context of this study, the relevant parameter to be
estimated here.
b. Report the 95% confidence interval for this parameter.
c. Does the 95% confidence interval calculated from the GSS sample data contain the
value 0? What does that imply? (Hint: Recall that a confidence interval is an interval
of plausible values for the parameter of interest, and the interval is calculated using the
sample statistics.)
d. Does the 95% confidence interval agree with your conclusion in #16? How are you deciding?
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e. Fill in the blanks of the sentence below to complete the interpretation of the 95% con-
fidence interval:
We are _______% confident that the mean number of close friends men have is
___________ than the mean number of close friends that women have,
by between ________ and _______ friends.
f. How would the width of the interval change if you increased the confidence level to
99%? Why?
g. How would the width of the interval change if you increased the sample size? Why?
h. How would the width of the interval change if the variability of the number of close
friends increased for both males and females? Why?
20. Step 6: Look back and ahead.
Looking back: Did anything about the design and conclusions of this study concern you?
Issues you may want to critique include:
• The match between the research question and the study design
• How the observational units were selected
• How the measurements were recorded
• The number of observational units in the study
• Whether what we observed is of practical value
Looking ahead: What should the researchers’ next steps be to fix the limitations in this
study and/or build on this knowledge?
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