Quantitative Methods
Assignment 1 worth 10%
Due: March 27 by 6:00 am ECT
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Assignments must be typed preferably. If handwritten solutions are submitted, please
ensure that they are legible.
Number your pages
Include a cover page with your ID Number
1.
A statistician selected a sample of 16 accounts receivable and determined the mean of the sample
to be $5,000 with a standard deviation of $400. She reported that the sample information
indicated the mean of the population ranges from $4,739.80 to $5,260.20. Based on the above
information, determine the level of confidence that was used.
[5 marks]
2.
A researcher was keen about the difference between men and women regarding how long they
take to complete a puzzle. The researcher calculated a 95% confidence interval for the
difference (in minutes) to be (-0.36, 0.25). Can she conclude that there is no difference?
[5 marks]
3.
A beer dispenser can be programmed so that it dispenses an average of µ millilitres (ml) per
glass. If the beer flow is normally distributed with a variance of 0.81 ml, determine the setting
for the dispenser so that 250 ml cups will overflow only 1 percent of the time.
[5 marks]
4.
The monthly income of Oistins residents is normally distributed with a mean of $3000 and a
standard deviation of $500.
(a) The mayor of Oistins makes $2,250 a month. What percentage of Whoville’s residents has
incomes that are more than the mayor’s?
[5 marks]
(b) Individuals with incomes of less than $1,985 per month are exempt from income taxes. What
percentage of residents is exempt from taxes?
[5 marks]
(c) What are the minimum and the maximum incomes of the middle 95% of the residents?
[5 marks]
(d) Two hundred residents have incomes of at least $4,440 per month. What is the population
of Oistins?
[5 marks]
5.
Market pioneers, i.e., companies that are among the first to develop a new product or service,
tend to have higher market shares than latecomers to the market. What accounts for this
advantage? Here is an excerpt from the conclusions of a study of a sample of 1209
manufacturers of industrial goods:
“Can patent protection explain pioneer share advantages? Only 21% of the pioneers claim a
significant benefit from either a product patent or a trade secret. Though their average share
is two percentage points higher than that of pioneers without a patent or trade secret, the
increase is not statistically significant (Z = 1.13). Therefore, at least in mature industrial
markets, product patents and trade secrets have little connection to pioneer share
advantages.”
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(a) Find the p-value for the given Z.
[5 marks]
(b) Explain to someone who knows no statistics what “not statistically significant” in the study's
conclusion means.
[5 marks]
(c) Why does the author conclude that patents and trade secrets don't help, even though they
contributed 2 percentage points to average market share?
[5 marks]
6.
A study shows that the amount of time spent by millennials playing video games is 22.6 hours
a month, with a standard deviation of 6.1 hours.
A bright UWI stats student has doubts about the study’s results. She believes that they actually
spend more time. The student tries to resolve her doubts, and collects a random sample of 60
millennials, asking them to keep a daily log of their video game playing habits. Millennials in
the sample played an average of 24.2 hours per month.
(a) If the null hypothesis is true, describe the sampling distribution of the mean number of hours
spent playing video games.
[5 marks]
(b) Calculate the probability of randomly choosing a sample in which the average number of
hours of video games played was 24.2 or more.
[5 marks]
(c) No hard and fast rule exists which divides the boundary between p-values for which we reject
the null and those for which we feel the null is plausible. However, p = 0.05 and p = 0.01 are
two commonly used thresholds. Under these thresholds, should the student reject the null
hypothesis?
[5 marks]
(d) Suppose the student doubted the study’s findings but had no prior expectation of whether
they were too high or too low. Perhaps she should determine the probability of randomly
choosing a sample in which the average number of hours spent playing video games was as
extreme or more extreme that 24.2 hours. Should she reject the null hypothesis in this case?
[5 marks]
(e) Would a larger sample with the same mean of 24.2 have provided stronger evidence of a
difference from the original study’s mean? Explain.
[5 marks]
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Case Study 1 (25 marks)
Standards Inc., a consulting firm, advises its clients about sampling and statistical procedures that can
be used to control their manufacturing processes. In one particular application, a client gave Standards
Inc a sample of 800 observations taken during a time in which that client’s process was operating
satisfactorily. The sample standard deviation for these data was 0.21; hence, with so much data, the
population standard deviation was assumed to be 0.21. Standards Inc then suggested that random
samples of size 30 be taken periodically to monitor the process on an ongoing basis. By analysing the
new samples, the client could quickly learn whether the process was operating satisfactorily. When the
process was not operating satisfactorily, corrective action could be taken to eliminate the problem.
The design specification indicated the mean for the process should be 12. The hypothesis test
suggested by Standards Inc is:
𝐻" : 𝜇 = 12
𝐻( : 𝜇 ≠ 12
Corrective action will be taken any time 𝐻" is rejected.
A number of samples were collected at hourly intervals during the first day of operation of the new
statistical process control procedure. These data are available in the data set Standards. [30 marks]
(a) Conduct a hypothesis test for each sample at the 0.01 level of significance and determine
what action, if any, should be taken.
(b) Compute the standard deviation for each of the four samples. Does the assumption of 0.21
for the population standard deviation appear reasonable?
(c) Compute limits for the sample mean around µ = 12 such that, as long as a new sample mean
is within those limits, the process will be considered to be operating satisfactorily. These limits
are referred to as upper and lower control limits for quality control purposes.
(d) Discuss the implications of changing the level of significance to a larger value. What mistake
or error could increase if the level of significance is increased?
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