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Statistical Analysis Homework #2
Notes:
• 2 points for each question.
• Each homework group only needs to submit one copy, via Blackboard
• You are strongly recommended to draw a curve/chart for each question.
• If you use z- or t- statistics, specify which one you are using.
• For hypothesis testing questions, clearly state null and alternative hypotheses.
• Clearly interpret your results.
• You can either use software of tables. If you use t-table and it does not have the exact df you are looking for, use the closest number instead.
1.
There are 36 students, each of whom submitted an examination paper. On the basis of years of
experience, the grader knows that the time needed to grade a randomly chosen examination
paper is a random variable with an expected value of 5 minutes and a standard deviation of 5
minutes.
a.) If grading times are independent and the grader begins grading at 5:50 P.M. and grades
continuously, what is the probability that he finishes grading before a 9:00 P.M. TV show?
b.) This year’s problems look quite different from previous years’, so you are unsure about the
average time for grading each paper. (Suppose the standard deviation for each paper is still 5
minutes). If in this year the grader finishes grading at 9:30 PM, construct a 95% confidence interval
for the population mean (for individual grading time).
c.) Based on information in b.) Do you think this year’s exam, in terms of grading time, is similar
to the previous years? Provide formal argument to support your statement.
2.
According to the Census ACS survey, the median household income for the United States was
$57,617 in 2016.
1 Suppose household income is normally distributed, with standard deviation
$25,000. You suspect that the average household income may have shifted in the past year (2017)
and decide to estimate it again using a random sample of 400 households. You remain confident
that the standard deviation of the income distribution has not changed.
a.) In constructing a confidence interval your friend suggests using the following instead of the
traditional one, which he thinks is boring. Consider the interval
How likely is it that this interval will contain the true population mean household income?
1 http://www.deptofnumbers.com/income/us/
b.) For 2017, you estimated a mean of $60,600 based on your sample. You wish to argue that the
US mean household income has indeed shifted (upward) compared to its 2016 level. What’s your
p-value? What conclusion will you get, at a significant level of 1%?
3.
Consider the following random sample of ten mpg (miles per gallon) numbers for a newly
introduced SUV:
20.5 22.3 26.7 16.8 19.2 28.2 14 23.2 19.8 24.6
Assuming that these were selected from a normal population distribution.
a.) Construct a 99% confidence interval for the population mean for mpg.
b.) The SUV has been advertised as having a mpg of at least 24. Based on your sample, do
you believe so? Let the significant level be 5%.
c.) Re-assess your answer in b.) at a significant level of 1%
4.
Recent information suggests that obesity is an increasing problem in America among all age groups.
Aiming to show that no more than 30% of adult Americans are obese, a study found that 963
individuals in a sample of 3325 adults are obese (a body mass index exceeding 30; this BMI index
is a measure of weight relative to height).
a.) Construct a 90% confidence interval for the population proportion of obesity.
b.) Does the study provide enough support to its goal, at a 5% significance level?
c.) If someone is designing a follow-up study for the same goal, but at a 1% significance level. Can
you help him to decide the minimal sample size needed (so that he knows how much budget is
needed)? [State any assumption(s) if needed]
5.
A new mini-course has been developed to enhance individuals’ financial competency. To evaluate
whether the course is effective, 416 subjects are invited to an experiment. Among those subjects,
200 subjects are randomly assigned to the treatment group, where they each takes the minicourse;
the rest (216) subjects are assigned to control group, where no mini-course is given.
At the end of the experiment, a test2 is given and the following results are reported:
mini-course Ave Score Sample SD n
Treatment Yes 3.62 1.6 200
Control No 3.30 1.8 216
Based on the experiment, do you think the mini-course is effectively enhancing subjects’ financial
competency? Why?