ONE AND TWO SAMPLE T-TESTS
Homework 7
Please include this sheet as a coversheet, fill it out and staple to your homework.
NAME: __________________________________________________
Class Time (OR “ONLINE”): _________________________________
Recitation Day/Time or TA (blank if online): _____________________
Grading:
Ex 1
Ex 2
Ex 3
Ex 4
Ex 5
Effort/neatness
8
8
8
12
12
TOTAL
2
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1. t-test of mean problem by hand The agricultural revolution in the 19th and 20th centuries is one of
the socially significant problems that spurred the development of statistics. One important example
is the discovery of the t-distribution.
William S. Gosset (pictured at right) was an Oxford-educated mathematician,
employed by the Guinness brewing company of Dublin. Since breweries
use lots of grain, they are naturally involved in agricultural research, so part
of Gosset’s work involved experimenting with crop yields. He realized that
new mathematical techniques for handling the resulting data were needed,
and he worked on the problem with Karl Pearson of University College,
London. The outcome of Gosset’s study was perhaps the most famous
paper in statistical literature, “The Probable Error of a Mean” (1908), which
introduced the t-distribution.
Since Gosset was contractually bound by Guinness, he published under the pseudonym,
“Student”, hence the t-distribution is often referred to as Student’s t-distribution. Here, we will
analyze one of the actual datasets that led to his analysis.
There is a reason to believe that drying seeds before planting them will increase the yield of the
crop. As part of his analysis, Gosset reported on the results of 11 trials in which crop yields were
obtained from ‘regular’ seeds as well as ‘dried’ seeds, and the change in yield (after drying the
seeds) was recorded.
Because few prior studies had been done, nothing was known about the standard deviation of
the population of all possible trials.
a. If drying the seeds before planting had no effect on crop yield one way or the other, what
should be the expected overall change in yield due to drying the seeds? [No calculation is
necessary here.]
b. The question of interest is whether drying the seeds before planting will increase the crop
yield. Please write the formal hypotheses for the test (your hypotheses should be stated in
symbols – with relation to a population parameter – not words).
Change in Crop Yield
Mean
33.72727273
Standard Error
19.95134578
Median
38
Mode
#N/A
Standard Deviation 66.17112801
Sample Variance
4378.618182
Kurtosis
-1.357191316
Skewness
-0.141741297
Range
197
Minimum
-70
Maximum
127
Sum
371
Count
11
Some summary statistics on the eleven data values are shown to the
left.
c.
From the output, report the value of the appropriate sample
statistic (use correct symbol to identify).
d.
To compute the standard deviation of the sampling distribtution
(the standard error), you will need one other value. In this case,
what is the symbol for the other value that you need, and what is its
meaning?
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e. Graphically represent a bell-shaped distribution of the appropriate statistic assuming the Null
Hypothesis is true. Label the mean of the distribution with the appropriate value; then mark
the value of the sample statistic in the appropriate location on the sketch, and shade the
associated ‘tail’ probability (in other words, draw your rejection region).
f. The value that you described in (d) can be found on the descriptive statistics shown above.
Do this, and then compute and report the standard error [show calculations].
g. Notice that Excel also computed the same result you just obtained in part (f). Did the value
that you calculated match what is listed in the output?
h. Calculate and report the test statistic, along with its appropriate symbol. Show all
calculations.
i. Use the table to find the critical value of the statistic at a significance level of 0.05 and state
your rejection region in terms of this value. Use the appropriate notation for the critical
value.
j. Decide and support your decision using the rejection region approach.
k. Conclude in terms of the problem. You should state the meaning of your decision in part (j)
in the context of the question regarding whether the experiment gives good evidence that
drying seeds before planting does, or does not, increase the crop yield.
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2. Companies that sell groceries over the internet are called e-grocers. Customers enter their orders,
pay by credit card, and receive delivery by truck. A potential e-grocer analyzed the market and
determined that the average order would have to exceed $85 if the e-grocer were to be profitable.
To determine whether an e-grocery would be profitable in one large city, she offered the service
and recorded the size of the order for a random sample of customers.
a. State the null and alternative hypotheses to determine whether and e-grocer will be
profitable in this city.
b. Choose the test statistic that will be used for this inference test, write the name of the test
that will be used.
The following Microsoft Excel output shows the result of the statistical test (from ‘Test
Statistics.xls’):
Sample mean
Sample standard deviation
Sample size
Hypothesized mean
Alpha
89.27
17.3
85
85
0.05
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
2.28
0.0127
1.6632
0.0254
1.9886
c. Using the information in the output above, create a sketch of this test. On your sketch, you
should have the following labeled with proper symbols and values: population mean,
sample mean, statistic, significance level (shaded), p-value (shaded).
d. Use the p-value approach to make a decision with respect to the hypotheses. Support your
decision with results from the output.
e. Write a 1-2 sentence conclusion in terms of the problem advising the potential e-grocer of
what she should do.
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[The following scenario/data will be used for problem 3]
One factor in low productivity is the amount of time wasted by workers. Wasted time includes time
spent cleaning up mistakes, waiting for more material and equipment, and performing any other
activity not related to production. In a project designed to examine the problem, an operationsmanagement consultant took a survey of 200 workers in companies that were classified as successful
(on the basis of their last annual profits) and another 200 workers from unsuccessful companies. The
amount of time (in hours) wasted during a standard 40-hour workweek was recorded for each worker.
3. Do these data provide enough evidence at the 1% significance level to infer that the amount of
time wasted in unsuccessful firms exceeds that of successful ones?
a. State the null and alternative hypotheses.
b. Based on your conclusion from the statistical test that you carried out in problem 3, what test
should be used to test the hypotheses that you put forth in part (a) [state the full name of the
statistical test]?
c. The sample data for the two groups was reported as follows:
Successful
Unsuccessful
5.02
7.80
s
1.39
3.09
n
200
200
Using the Test Statistics workbook in Excel (file is on Courseweb under “Course
Documents”), conduct the test that you have listed in part (b). Attach your results.
d. Using your results from part (c), make a decision with respect to your null and alternative
hypotheses. Support your decisions with values from your output.
e. List the conditions that must be met in order to use this test – how would you go about
testing them?
f. Write 2-3 sentences that conclude the test that you carried out in problems 3 and 4 that
could be used by the operations-management consultant in his report to an unsuccessful
company.
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4. Use computer software to test the hypotheses that you stated in Homework 6, part 6c. You may
choose the significance level at which you will carry out your test (make sure that it’s noted
somewhere). You should carry out the following steps (as we have practiced in class and on all of
the hypothesis testing problems) and attach to your assignment:
a. State hypotheses
b. Choose the proper statistical test to be used and state what information helped you to come
to this decision. Check any conditions that you may have to meet in order to carry out your
test (show any related output). Perform the test using Microsoft Excel (using the Test
Statistics workbook) and attach your output.
c. State your decision and support it with values from the output.
d. Conclude in terms of your topic.
5. Answer the following questions about the dataset that you’re analyzing this term:
a. List the categorical (nominal or ordinal) variables that you have:
b. Try to find a statement that someone has made about one of these categorical variables
that you have listed in part (a). For example, if I chose to analyze student satisfaction with
the current textbook, I may use a statement made by the the publisher that “The majority of
business statistics love this book.” If you cannot find a statement online, you can ask
someone about the topic (that is, ask the question “what do you think about the business
statistics book you’re using? Do you (1) LOVE IT, (2) don’t care, (3) hate it”) and use this
as your value. Copy or write the statement below:
c. Make a claim. What do you think about this claim? Do you think that the true value will
actually be higher, lower, or different (that is, you’re not sure the direction of the difference)?
d. Write null and alternative hypotheses putting your answers in parts a and b in the form of a
hypothesis about the true population proportion p. [use appropriate symbols]
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