**Stat 301 – HW 3**

**Due midnight, Friday, Jan. 27**

*Please
remember to put your name(s) inside the file and if submitting jointly to join
a HW 3 group first. Please use Word or PDF format only. Remember to integrate your output with your
discussion. Points will be deducted if
you are missing output. *

Due to injury
or other medical conditions, people can develop blindness on just one side of
their vision field. “Blindsight” is a condition in which people are blind but
can still respond to things they cannot consciously see. Because of brain
damage, a patient whom researchers called GY (Persaud, et al., *Nature Neuroscience*, 2007) was right-side blind, meaning he could not see anything in his
right field of vision. To test GY for
blindsight, they had him face a video monitor and told him that “either a square-wave horizontal
grating pattern would be presented in his right visual field (within his
scotoma) or no stimulus would be presented. The size, contrast, and position of
the stimuli were selected on the basis of previous experiments with GY, so that
he would be expected to make present-absent decisions with an accuracy above
chance, but would report no visual sensation. He was told that pattern and
blank trials would be equally frequent. … He was asked to respond 'yes' if he
thought the stimulus had been presented and 'no' if he thought it had
not.” In a set of trials in his blind
field, GY correctly made 70% of the yes-no discriminations.

(a) Identify the observational units and
variable of interest in this study.

(b) Is it reasonable to model this
process as binomial (in other words, how did they/would you try to ensure
independence and constant probability of success for this random process)?

(c) Define the parameter in words (in
context).

(d) Report the sample proportion, .

We can’t decide whether this result is
better than random chance without knowing the sample size.

(e) Suppose the sample size had been *n*
= 10. Use the One Proportion Inference applet
to find the one-sided binomial p-value. Make sure you include a copy of the
null distribution *for the sample proportion *with the p-value shaded and
showing the mean and standard deviation. (Recommendation: Verify the SD
calculation by hand.)

(f) Check the Normal Approximation box
and report the theory-based p-value for the one-sample *z*-test. Would you consider the p-values similar? Does
the similarity/lack of similarity of these values surprise you? Explain.

(g) The actual sample size was *n*
= 200. Find the one-sided binomial
p-value. Make sure you include a copy of the null distribution *for the
sample proportion *with the p-value shaded, and showing the mean and
standard deviation. Is this p-value larger or smaller than in (e)? Is this what
you expected for this change in sample size? Explain.

(h) Compare the shape, center, and
spread of the two null distributions you created in (e) and (g). Which features
are the same/which are different with the change in sample size?

(i) Describe another way the two
distributions differ (visually).

(j) For *n* = 200, calculate the
standardized statistic for the sample proportion (the “test statistic”). (Show your work.) Then compare the exact
binomial p-value to the normal approximation p-value. Would you consider the p-values similar?

(k) Ok, that was a silly question. Suppose *n *= 200 and =
0.55. Find the exact binomial and theory-based p-values. Are they similar? Does the similarity/lack of similarity
of these values surprise you? Explain.

(l) The *continuity correction* is
discussed on p. 58-59. Use the applet to
perform a continuity correction for the calculations in (k). Does this continuity correction improve the
normal approximation of the exact binomial calculation for this situation?
(Make sure you are including sufficient output.)

From the 200 trials ( =
0.70), we have strong evidence that GY’s probability is larger than 0.50. So what is GY’s actual (long-run) probability
of correctly answering? This is what a confidence interval tells us/helps us
estimate.

(m) Use R or JMP to find the exact
binomial confidence interval. Interpret
the interval in context. (If I don’t give you a confidence level, assume 95%.)

(n) Repeat (m) for a 95% *z*-confidence
interval. Are the
*z*-interval and exact binomial confidence intervals similar? Is
this what you would expect for these data? Explain.

(*We can discuss on Friday) *

(o) Repeat (m) for a 95% adjusted-Wald
confidence interval. Make sure it’s
clear how you are doing so.

(p) Compare the widths of the three
confidence intervals you have found. (Use 4 decimal
places.) Which is the shortest?

(q) Consider the one-proportion *z*-confidence
interval in (n). Describe (verify?) how the interval will differ for *n* =
200 and =
0.55 (consider the midpoint and the width). Explain.

(r) Consider the one-proportion *z*-confidence
interval in (n). Describe (verify?) how the interval will differ for *n* =
10 and =
0.70 (consider the midpoint and the width). Explain.

(s) Report the half-width (aka *margin
of error*) for the one-proportion *z*-confidence interval in (n). Compare this value to . Based on the formula for the margin of
error in the 95% one-proportion *z*-confidence interval, why does this
approximation make sense?

(t) So a short-cut approximation to the
one-proportion *z*-confidence interval is .
If anything, this interval will be wider than it needs to be (is
“conservative”). Why? (*Hint*:* *How does the standard
deviation formula change with ?)