STAT2203: Probability Models and Data Analysis for Engineering
Assignment 2
Due by 14:00 on Friday the 30th of October, 2020. Submission via Blackboard.
The marks for each question are indicate by the number in square brackets. There are a
total of 15 marks for this assignment.
1. A drug was tested to determine whether it could lower the serum thyroxine levels of
patients with Graves’ disease. 32 patients with Graves’ disease participated in the
study, and were allocated randomly to receive either the drug or a placebo over an
eight-week period. There were 15 participants in the drug group, and by the end
of the study, this group’s sample mean serum thyroxine level had fallen by 21%,
with a standard deviation of 27%. There were 17 patients in the placebo group.
The placebo group’s sample mean serum thyroxine level decreased by 1%, with a
standard deviation of 24%.
(a) Give a 95% confidence interval on the mean percentage decrease in serum
thyroxine level for patients that would receive the drug. [1 mark]
(b) Does the drug lower the mean serum thyroxine level more than the placebo?
State the null and alternative hypotheses, compute the appropriate test statis-
tic, and compute the p-value. What do you conclude? [2
marks]
2. A survey of the Australian population was conducted in 2013 concerning participa-
tion in sport. Of the 125 respondents from Queensland 69 reported that they had
participated in sport at least once during the 12 months prior to the survey.
(a) Construct a 99% confidence interval for the proportion of Queenslanders that
participated in sport during this period. [1 mark]
(b) Of the 84 respondents from Tasmania 56 reported that they had participated
in sport at least once during the 12 months prior to the survey. Is there
any evidence of a difference in the proportion of people participating in sport
between the two states? State the null and alternative hypotheses, compute the
appropriate test statistic, and compute the p-value. What do you conclude?
[2 marks]
3. Data from the OECD suggest that Australians have a below-average work-life bal-
ance when compared to other OECD countries. One question of the General Social
Survey conducted by the ABS asked how often people felt rushed or pressed for
time. The data below is a representative sample from one suburb.
1
Feeling rushed or pressed for time Males Females
Always / Often 36 45
Sometimes 29 35
Rarely / Never 36 19
Based on the table above, is there any evidence of an association between Gender
and ‘Feeling rushed or pressed for time’? State the null and alternative hypotheses,
compute the appropriate test statistic, and compute the p-value. What do you
conclude? [3 marks]
4. Answer the following questions with the help of MATLAB or R. In MATLAB you
might find the following functions useful: binocdf, binopdf, binoinv. In R you
might find the following functions useful: pbinom, qbinom, dbinom. Read the help
for these functions to ensure you are using them correctly.
(a) After rolling a die a few times you become suspicious that 6’s are occurring
too frequently compared to a fair die. You are prepared to roll the die 30 times
and record the number of 6’s that occur. How many 6’s would you need to
observe for the p-value from the appropriate hypothesis test to be less than
0.05? [1 mark]
(b) Let Tc denote the smallest number of 6’s would you need to observe for the
p-value from the appropriate hypothesis test to be less than 0.05 (the answer
from part a). Let pd denote the probability of getting a 6 after rolling this
dice. Plot the probability of observing Tc or more 6’s from 30 rolls against pd.
From the plot determine the smallest value of pd such that the probability of
rejecting the null hypothesis at the 5% significance level is at least 0.8. [1
mark]
(c) Suppose pd is in fact 1/4. Plot the probability of observing Tc or more 6’s from
N rolls against N . From the plot determine the smallest number of rolls such
that the probability of rejecting the null hypothesis at the 5% significance level
is at least 0.8. [1 mark]
5. Suppose the random variable X has a Uniform(0,1) distribution. Conditional on
{X = x}, the random variable Y has a N(−x, x2) distribution. Using the property
that E[U ] = E[E[U |V ]] or otherwise, determine E[Y ] and Var(Y ). [3 marks]
Total [15 marks]
2

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