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The answers for exam 3 of the stat 205 course, which covers topics such as p-values, hypothesis testing using t-distributions and chi-square tests, confidence intervals, and significance tests for means and proportions.
Typology: Exams
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Part I: Answer eight of the following nine questions. If you complete more than eight, I will
grade only the first eight. Five points each.
The P-value of a test is the probability under H (^) 0, of observing a test statistic as extreme or
more in the direction of H (^) A as that actually observed.
differences come from a normal population. We use a t distribution hypothesis test rather
than a sign test because the t test is more powerful than the sign test.
trait in humans. In Europe and Asia, about 70% of people are “tasters”. Suppose a study is
being conducted to estimate the population proportion of tasters (via a 95% confidence
interval). The researchers would like to keep the margin of error for the confidence interval
less than or equal to 0.01. Write down the formula – with the appropriate numbers in it (no
letters, please) – that will tell the researchers what sample size to take in order to meet this
criterion. You do not need to simplify this expression.
2 2
2
n ≥
of error, we should round down / round up to the next integer value.
Part II: Answer every part of the next two problems. Read each question carefully,
and show your work for full credit.
excretion (grams of protein per gram of creatinine) was measured for each patient
before and after plasmapheresis. The data and QQplot of the differences are given
below.
Patient Before After Difference
Mean 8.9 0.9 7.
1a) (10 points) Use the QQplot of the differences above to comment on whether the
assumption of normality has been met. Two or three sentences should suffice.
There is a systematic departure from the line. The “U” shape is indicative of a skewed
right distribution. This is a clear indication these data do not come from a normal
population and with only 6 data points, we cannot invoke the CLT. The normality
assumption has not been met.
1b) (25 points) Conduct a sign test to investigate at the 0.05 significance level whether or
not urinary protein excretion tends to go down after plasmapheresis in patients with renal
disease.
(1) α = 0.
(2) H (^) 0: Urinary protein excretion is the same before and after plasmapheresis
HA: Urinary protein excretion tends to go down after plasmapheresis
(3) H (^) A: “effect before” > “effect after”, then B (^) s = N+ = 6
(4) P = Pr{Bin(6,1/2) ≥ 6} = Pr{Bin(6,1/2) = 6} = 6 C6(.5)
6 (.5)
0 = 0.
6 = 0.
Or, using the TI calculator
P = 1 – Pr{Bin(6,1/2) ≤ 5} = 1 – binomcdf(6, 0.5, 5) = 0.
(5) P < α, reject H 0
(6) We have significant evidence to conclude urinary protein excretion tends to go
down after plasmapheresis.
