**Stat 301 –
HW 5**

**Due Friday,
Feb. 17, midnight**

**These questions focus on Ch. 2. (Meaning: start now!) There are definitely
some “stretch” questions so try your best.
Please include your name(s) in the file and include all relevant output.**

**1) **Researchers at Arizona State University (McNabb
& Gray, 2016) explored the effects on driving with various types of
cell phone use. In particular, they were interested in comparing the effects
between text-based media and picture-based media. They had their subjects use a
driving simulator and requested that they stay two seconds behind the car in
front of them. The car in front travelled between 55 and 65 mph and was
programmed to come to a complete stop 8 times during the simulation. One of the
variables the researchers measured was the reaction time for the subjects to
brake when the car in front of them stopped. Each subject was measured twice:
once when instructed to scroll through Facebook messages that just consisted of
text and once when instructed to scroll through Instagram pictures that did not
contain any text. Datafile: BrakeReactionTime.txt

(a) The measurement units were not
given, what do you think they were? How did you decide?

(b) In the study, the order of the two
treatments was randomly determined for each subject. Why do you think that was?

(c) Calculate the difference in reaction
time (Facebook – Instagram). Create a graph of the distribution of the
differences. Write a sentence
summarizing the behavior of these differences, including what preliminary
evidence is provided for the research question.

(d) Let represent the long-run mean difference in
reaction times (Facebook – Instagram). Write appropriate null and alternative
hypotheses for .

(e) Are the one-sample *t*-procedures
likely valid for these data? Explain your reasoning.

(f) Use R or JMP or Theory-Based
Inference applet (p. 147) to carry out a one-sample *t*-test for the
differences (regardless of your answer to (e)).
Include your output **and **summarize your conclusion in context.

(g) Use R or JMP or Theory-Based
Inference applet to determine a 95% *t*-confidence interval. Include your output **and** write a
one-sentence interpretation of your interval.

(h) Determine **and interpret** a 95%
prediction interval. Be sure to show details of how you determined the
interval.

**2) **An alternative to a one-sample *t*-procedure is *bootstrapping*.
Bootstrapping is most helpful when the *t*-procedures are not expected to
be valid, especially when the choice of statistic is not the mean. In
particular, bootstrapping can provide an estimate of the standard error of the
statistic without using the theoretical formula or .
The principle behind bootstrapping is to estimate “sample to sample”
variation in the statistic by taking repeated samples from the sample you have,
but with replacement. (Note: It’s
important to match the sample size of the study. Sampling *with replacement *is
what allows the results to differ from sample to sample. See also Investigation
2.9. You can think of this as taking samples of 18
observations from a population that consists of infinitely copies of your original
sample.) Below is a bootstrap distribution of the sample median for the
reaction time data.

resamples =
lapply(1:1000, function(i) sample(diffs$differences, 18, replace = TRUE) )

bootstrapmedians
= sapply(resamples, median)

Figures: 1,000 sample medians from 1,000
random samples (*n* = 18) with replacement from the observed reaction time
differences

(a) Try to explain why the mean of this
bootstrap distribution is closer to 0.19 than to 0 based on the bootstrapping
process.

(b) There appears to be several clumps of
identical values in the dotplot. Try to
explain why that is not surprising for a graph of sample *medians*.

(c) What is the standard deviation of the
bootstrap distribution? Use this value
to create an informal 95% confidence interval for the population median (e.g.,
estimate __+__ 2 SE(estimate)).
Include a one-sentence interpretation of your interval in context.

(d) How does the interval for the
population median compare to the interval for the population mean? Which
interval would you report to these researchers? Why?

**3)** Continuing with the reaction time data.

(a) Reporting “effect size” is becoming
increasingly popular. For a one-sample *t*-test,
the effect size could be calculated as .
How does this differ from a one-sample *t*-statistic and conjecture
why knowing the effect size of a study might be useful (*Hint*: practical
significance).

(b) Calculate the effect size for the
reaction time data. (This is also called Cohen’s *d*, and Cohen in the 90s
decided 0.5 was a moderate effect (= “visible to the naked eye of a careful
observer”) and 0.8 was a large effect. Related article.

(c) Explain what a “Type I Error” would
mean in this context.

(d) Explain what a “Type II Error” would
mean in this context.

(e) Suppose the researchers considered a
mean difference of 0.33 seconds (the length of an eye blink) to be
meaningful. Use R or JMP to determine
the power of the one-sample *t*-test to detect this difference.

#You have to first install a package install.packages("pwr") #Then load in the package library("pwr") #Now use the pwr.t.test function pwr.t.test(n = 18, d = 0.33/.41,
sig.level = 0.05, type = c("one.sample")) Explain what d is here. |
Choose Keep Alpha set to 0.05. Change the value of the Std Dev to
0.41. Specify 0.33 as the difference to
detect. Enter 18 as the sample size. |

Include your output.

Report and interpret the power in
context. Does this amount of appear
sufficient to you?

(f) Repeat for a sample size of *n*
= 100. Include your output. Does the
power increase or decrease?

(g) Try to give an intuitive explanation
for why the power in (e) is so large even though the sample size is so small.

(h) In addition to impacting power, what
is a second advantage to using a larger sample size for a one-sample *t*-test?