**Stat 301 - HW 8**

**Due midnight, Friday, March 17**

*Remember to
join a HW group if you are submitting together, to put your names in this file
and to include, and to integrate all relevant computer output. *

**1)** Stat
301 students (Coutin & Heffler, 2021) wanted to know whether listening to
up-tempo music causes college students to tend to type faster. To collect their
data (number of words typed correctly in one minute), the students planned to
use the 60-second Easy-Text typing test (TypingTest.com).
They recruited 34 Cal Poly students from groups they were associated with on
campus (e.g., athletic teams, musical groups).
For the up-tempo music they selected Overture to Candide performed by
the London Symphony Orchestra.

Let μ_{nomusic}
represent the population mean typing speed without the music and μ_{music} the population mean typing seed with
the music.

(a) State the student researchers’ null and alternative
hypotheses in symbols **and** in words.

(b) Briefly describe what a “completely randomized design”
(independent samples) would look like. (Be clear how randomization is used.)

(c) Briefly describe/contrast what a matched pairs design
would look like. (Be clear how randomization is used.)

(d)
According to typingpal.com, an average typing speed is 40 words per minute, and
a good targe speed is 65 to 70 words per minute. Suppose we will be impressed if listening to
music increases typing speed by 5 words per minute on average. Also suppose the variability (standard
deviation) of typing speeds is around 14 words per minute. (Note, this corresponds to an effect size of
5/14 = 0.36.) No question here!

(e) Suppose we plan to use a completely randomized design,
randomly assigning 17 students to each treatment. Find the power we would detect a difference
in population mean typing speed in a two-sided test. We had a direction in mind
above, but two-sided is the default in both R and JMP, so let’s start there.

R power.t.test(n=17,delta=5,sd=14,type="two.sample") |
JMP Choose Choose Enter 14 as the standard deviation Enter 5 as the difference to detect Enter 34 as the overall sample size (Leave power blank) Press Continue (JMP then fills
in the power) |

Include your output and give a one-sentence interpretation of
the calculated power in the context of this study.

(f) Suppose we plan to use a matched pairs design, asking
each student to take the test twice, once with music and once without,
randomizing which method they use first.
Should the variability in the *speed differences* be larger or
smaller than the variability in the *speeds*? Explain your reasoning
(should be clear how you are contrasting the two).

(g)* *Suppose the variability in the *speed
differences* is 10 words per minute.
Calculate the power that we would detect a nonzero mean difference in
typing speed in a two-sided test.

R power.t.test(n=17,delta=5,sd=10,type="paired") or power.t.test(n=34,delta=5,sd=10,type="paired") |
JMP Choose Choose Enter 10 as the standard deviation Enter 5 as the difference to detect Enter 17 or 34 as the sample size Press Continue |

Include your output. How do the power calculations compare?

The data in TypingMusic.txt shows the speeds (in words per minute) for each
student using music and not using music.

(h) Create a graph of the differences. Include your output
and summarize what you learn. (JMP users, you may want to calculate the
differences in Excel?) Also, calculate
the standard deviations of each group (with and without music) as well as the
standard deviation of the differences (include output). Is the standard
deviation of the differences a lot smaller? (If so, then that says the pairing
was helpful.)

(i) Use R or JMP to carry out the ** one-sample t-test**
on the differences (aka a matched-pairs

R library(readr) typing <- read_delim("http://www.rossmanchance.com/iscam3/data/TypingMusic.txt",
delim = "\t") t.test(typing$WithMusic, typing$NoMusic,
paired=TRUE) |
JMP To run a matched pairs analysis using
the original two columns of data:
Enter both columns in the Y, Paired
Responses box and press OK. |

Include a screen capture of the results and report the test
statistic and two-sided p-value.

(j) Does the *t*-test appear to be valid for these
data? You should comment on the validity conditions of the paired *t*-test.
(Probability plot?)

(k) Regardless of your answer to (j), use R or JMP to
calculate **and interpret** a one-sample *t*-confidence interval in
context (including measurement units).

(l) Carry out a ** sign test** on the paired data:

1. How
many of the difference are positive? How many are negative? How many are zero?

2. Consider
the non-zero differences (i.e., throw-away the ties), what proportion of the
non-zero differences are positive?

3. Use
the binomial distribution to determine whether a statistically significant
majority of the differences are positive (define the parameter of interest,
state the hypotheses, and determine the exact binomial p-value – be sure to
include a screen capture of the null distribution showing the p-value).

(m) Does the sign test provide stronger or weaker evidence
that one typing method tends to be faster than
the other? (Be *very *clear how you are deciding.)

(n) Would a one-sample *z*-test be appropriate in
(l)? Explain how you are deciding.

(o) Determine, include output, and interpret in context a
95% confidence interval for . This time, consider your answer to (n) in
deciding which interval procedure to use.

**Reminder:** Course
Evaluations will be due Friday night as well!