MATH08051: Statistics (Year 2)
Assessment 3
2021-2022
Please answer all questions in full. Where appropriate, add text within your calculations/derivation
to help explain your reasoning at each step. Upload images of your scripts to Gradescope be-
fore the deadline (see below). Tag each page of you submission to the appropriate question
(see Gradescope instructions for more details.) Each question is graded out of 5 based on the
accuracy and quality of your submission.
DEADLINE: 08 March 2022 at 16:00 UK local time.
1. Read the following statement:
“We have conducted a two-sided Z-test to assess the null hypothesis that the mean con-
tents of soda cans is 330ml. The p-value of the test is 0.74, meaning that the probability
of the null hypothesis statement being correct is 74%. As the p-value is much greater
than the specified 5% type II error rate, it follows that the true mean soda can contents
is indeed 330ml.”
There are a number of inaccuracies in how the author has reported the findings of their
statistical test. Identify and briefly explain the mistakes.
2. Let X be exponentially distributed with population mean µ. Recall that the exponential
cdf is F (x) = 1− exp(−xµ) for x > 0. Suppose that we wish to test the hypothesis:
H0 : µ = 1 vs H1 : µ < 1.
We define the test statistic to be the value of a single random variable, X. From using
a 5% significance level, show that the critical region for the test statistic is given by
C = (0,− log 1920). Given this, derive the power function, β(µ), for the hypothesis test.
3. A study is conducted to assess if doing puzzles has any impact on heart rate. Eight
volunteers had their at-rest pulse rate measured before and after the experiment. The
differences (rate after - rate before) are tabulated below:
Individual 1 2 3 4 5 6 7 8
Difference -3 -5 1 0 -4 3 -7 -1
Let µ denote the difference in population means (“after - before”). Conduct a two-
sided paired t-test at the 5% significance level. Clearly state the null and alternative
hypotheses, and provide an appropriate concluding statement explaining your results.
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