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ECO252 QBA2
FIRST EXAM
February 28, 2008
Version 1
Name ________________
Class hour: _____________
Student number: __________
Show your work! Make Diagrams! Include a vertical line in the middle! Exam is normed on 50 points. Answers without reasons are not usually acceptable.
I. (8 points) Do all the following. x ~ N

7 , 11

(But you can’t buy donuts there!)
1.
P
 x

0

2. P

14
 x

42

3. P


30
 x

30

4. x
.
0005
(Do not try to use the t table to get this.) (I only need one answer – you may find more than one possibility.)

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II. (9 points-2 point penalty for not trying part a.)
Langley) A copier has been turning out 45 copies a minute. After a repair, the copier is tested 6 times. In these 5 runs the output per minute is 46, 47, 48, 47, 47, 46 a. Treating the above numbers as a random sample of size 6 from the Normal distribution, compute the sample standard deviation, s , of expenditures. Show your work! (2) b. Compute a 90% confidence interval for the mean output per minute. (2) c. Redo b) when you find out that there were only 20 runs to pick the sample of 6 from. (2) d. Assume that the population standard deviation is 0.7 and create a 99.9% two-sided confidence interval for the mean. (2) e. Use your results in a) to test the hypothesis that the mean is above 45 at the 90% level. (3) State your null and alternative hypotheses clearly! f. (Extra Credit) Test the hypothesis that the population standard deviation is 0.70 at the 99.9% significance level assuming that a random sample of 50 yielded a sample standard deviation of
0.75. (3)

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III. Do as many of the following problems as you can.(2 points each unless marked otherwise adding to
13+ points). Show your work except in multiple choice questions. (Actually – it doesn’t hurt there either.) If the answer is ‘None of the above,’ put in the correct answer if possible. ( ) gives points for the question. [ ] gives a running total.
1) If I want to test to see if the population mean of x is smaller than 5 my null hypothesis is: i)
 
5 ii)
 
5 iii) iv)
 
5
 
5 vi) None of the above. (So what is it?) vii) Any of i)-iv) could be right. We need more information.
2) Assuming that you have a sample mean of 100 based on a sample of 36 taken from a population of 900 and you are testing to see if the population mean is 90 with a known population standard deviation of 80, the 99% critical values for the sample mean are a) 100

2 .
576


80
36

 b) 100

2 .
626

 c) d) e) f)
100
90
90
90




2
2
2
2
.
.
.
576
576
626
.
576








80
36


80
900


80
36


80
36


80
900


3) Which of the following is a Type 1 error? a) Rejecting the null hypothesis when the null hypothesis is false. b) Rejecting the null hypothesis when the null hypothesis is true. c) Not rejecting the null hypothesis when the null hypothesis is true. d) Not rejecting the null hypothesis when the null hypothesis is false. e) All of the above f) None of the above.

252x0811 2/20/08 (Open in ‘Print Layout’ format)
4) (Langley) It is generally believed that 15% of white Australians are allergic to penicillin. A doctor believes that the allergy occurs in a lower proportion of Native Australians. To test that belief a random sample is gathered of 50 Native Australians and it is found that only one (2% of the sample) is allergic to penicillin. The doctor creates a p-value to compare against a significance level of 5%. What do we mean by a p-value? [8] a) P-value is the 2% proportion. b) P-value is c) P-value is d) P-value is e) P-value is f) P-value is g) P-value is h) P-value is i) P-value is j) P-value is
2
2
P
P
2
2
P

P
P
P


P
 p


 p p

 p p p p







.
.
02
02
.
02
.
02
.
15


.
.
15
.
05




.
.


.
.
P
P p p

2

.
15
.
15
.
.

.
.
Exhibit 1: (Langley) Langley’s daughter loved to play chutes and ladders. She told her daddy, however, that the one die she used seemed to come up a six when she didn’t want a six. A fair die is equally likely to come up with each of its six faces on top. Langley now suspected that the die was coming up a six more often than it should. As an experiment, he rolled the die 108 times and it came up a six 25 times.
5) We wish to test whether Langley’s suspicion in exhibit 1 is correct. To do so, we must do which of the following: [10] a) A z-test of the population mean. b) A z-test of a population proportion. c) A t-test of the population mean. d) A
 2
-test of the population variance. e) A test of the population median f) None of the above (To get full credit propose a test type.)
6) In Exhibit 1, what are the null and alternative hypotheses that Langley should be testing. (2)
7) In Exhibit 1, what is the value of the test ratio that you would use to test your hypotheses in 6)? Show your work. (Note that this could be right even if the answer to 6) is wrong.) (3)
8) Using a 95% confidence level, explain, using your hypotheses, whether the die was fair. (2) [17]

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Exhibit 2: (Ng) The manager of the credit department believes that the average balance held by credit card holders is $75. A random sample of 29 accounts is selected and she finds that the sample mean of the amount owed is $83.40 and the sample standard deviation is $23.40. It is believed that the distribution of the population is approximately Normal.
9) We wish to test whether the manager’s belief in exhibit 2 is correct. To do so, we must do which of the following: (1) [18] a) A z-test of the population mean. b) A z-test of a population proportion. c) A t-test of the population mean. d) A
 2
-test of the population variance. e) A test of the population median. f) None of the above (To get full credit propose a test type.)
10) a) State your null and alternative hypotheses to test the manager’s belief in Exhibit 2 (1). b) Give an appropriate critical value or values (for a mean, proportion, variance or median). (2) [21]
11) The manager of the credit department believes that the median balance held by credit card holders is above $75 and that the population does not have a Normal distribution. A random sample of 100 accounts is selected and 60 of the accounts have balances above $75. Which of the following is the null hypothesis that the manager will end up testing? (To protect yourself, you might want to explain what p is. Otherwise
I will use my own assumption.) (2) [23] a) b) c) d) e) p

.
5 p

.
5 p

.
5 . p

.
5 p

.
5 f) None of the above (To get full credit propose a null hypothesis.)

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Blank page for calculations .

252x0811 2/20/08 (Open in ‘Print Layout’ format)
ECO252 QBA2
FIRST EXAM
February 28, 2008
TAKE HOME SECTION
-
Name: _________________________
Student Number and class time: _________________________
IV. Do at least 3 problems (at least 7 each) (or do sections adding to at least 20 points - Anything extra you do helps, and grades wrap around) . Show your work! State H
0
and H
1
where appropriate. You have not done a hypothesis test unless you have stated your hypotheses, run the numbers and stated your conclusion.
(Use a 95% confidence level unless another level is specified.) Answers without reasons usually are not acceptable. Neatness and clarity of explanation are expected. This must be turned in when you take the in-class exam. Note that answers without reasons and citation of appropriate statistical tests receive no credit.
Failing to be transparent about which section of which problem you are doing can lose you credit. Many answers require a statistical test, that is, stating or implying a hypothesis and showing why it is true or false by citing a table value or a p-value. If you haven’t done it lately, take a fast look at ECO 252 - Things That You Should Never Do on a Statistics Exam (or
Anywhere Else) .
Problem 1: (Doane and Seward) A fast food restaurant has just started serving hot cocoa. The management wishes to serve cocoa of an average temperature of 142 degrees. 24 measurements of the temperature in 10 stores are taken. You are manager of store a and will use the corresponding column, where a is the second to last digit of your student number. (For example, Seymour Butz’ student number is 543987 so he uses column x8.) If that number is zero, use column 10. You are testing to see if the mean for your store is
142. There will be a penalty if you do not make it clear what column you are using.
Row x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
1 140 142 142 143 144 142 143 146 143 143
2 142 143 143 138 139 142 144 145 144 139
3 141 141 141 140 144 142 144 144 141 145
4 142 142 142 140 143 140 145 144 144 144
5 141 139 142 142 141 141 146 145 142 145
6 141 142 140 139 142 141 142 144 140 142
7 145 144 143 142 141 137 145 141 142 141
8 142 145 142 139 138 142 142 141 141 140
9 142 143 141 145 144 139 145 144 146 142
10 142 143 136 139 145 141 143 144 140 142
11 137 141 142 142 139 141 142 139 143 142
12 139 139 138 138 141 143 142 142 144 146
13 139 143 142 142 145 141 141 141 142 142
14 144 144 139 141 142 142 147 142 143 143
15 140 140 140 142 144 140 141 144 142 142
16 141 138 143 141 145 142 137 145 141 140
17 140 141 139 141 142 142 142 139 142 144
18 140 140 139 140 144 142 140 142 144 143
19 140 142 139 142 136 139 144 143 144 141
20 139 140 141 141 138 142 142 146 145 144
21 146 138 143 143 141 143 147 142 145 143
22 138 139 141 141 142 143 146 144 141 141
23 139 142 140 140 140 142 140 144 143 139
24 142 141 143 140 141 140 143 146 142 142
Assume that the Normal distribution applies to the data and use a 98% confidence level. a. Find the sample mean and sample standard deviation of the incomes in your data, showing
your work. (1) (Your mean should be between 140 and 146 and your sample standard deviation should be around 2.) b. State your null and alternative hypotheses (1) c. Test the hypothesis using a test ratio (1) d. Test the hypothesis using a critical value for a sample mean. (1)

252x0811 2/20/08 (Open in ‘Print Layout’ format) e. Test the hypothesis using a confidence interval (1) f. Find an approximate p-value for the null hypothesis. (1) g. On the basis of your tests, is the mean temperature correct in your restaurant?? Why? (1) h. How do your conclusions change if the random sample of 24 temperatures is taken on a day in which only 48 cups cocoa are sold? (2) i. Assume that the Normal distribution does not apply and test to see if the median is 142. Be careful! What should you do with numbers that are exactly 142? (2) j. (Extra Credit) Do a 98% confidence interval for the median. (2)
[12]
Problem 2: Once again assume that the Normal distribution applies to the data in Problem 1, but that we know that the population standard deviation is 2. Our confidence level remains 98%, but we are now testing the hypothesis that the mean is below 143 degrees. a. State your null and your alternative hypotheses. (1) b. Find the value of
98%. z that you need for a critical value for a 1-sided test if the confidence level is c. Find a critical value for the sample mean to test if the mean is below 143 degrees. (1) d. Test the hypothesis that the mean is below 143 degrees using an appropriate confidence interval. (2) e. Using your critical value from 2b, create a power curve for your test. (6) f. Assume that the population standard deviation is 2. How large a sample do you need to get a two-sided 98% confidence interval with an error not exceeding

0 .
5 degrees? (2) [22]
Problem 3: According to Doane and Seward about 13% of goods bought at a department store are returned. An organization called Return Exchange will sell you a software product called Verify-1for which it makes the claims below.
Verify-1® is quickly operational. And it authorizes returns even quicker
Verify-1® identifies fraud and abuse at the point of return before they become liabilities to your brand equity or profits. In stand-alone mode, this easy-to-use, turnkey solution can be operational in 30 days and will reduce your return rate immediately, without disrupting your business or IT configuration. Verify-1® also integrates easily into your existing POS platform.
You set the policy, Verify-1® enforces it
With Verify-1®, your returns are dealt with consistently utilizing advanced statistical modeling in combination with state return laws and your existing return policies. At the point of return, using the customer’s driver’s license or other valid identification, Verify-1® automatically checks prior return behavior and authorizes or declines the transaction. Customers identified as risks for presenting fraudulent returns are declined, while legitimate returns are speedily accepted.
You take a sample of n items and find that there were x returns (about 9%).You are the manager of store a . ( a is the last digit of your student number. (For example, Seymour Butz’ student number is 543987 so he manages store 7.) The sample size and number of returns for your store is given below. On the basis of this sample, can you now say that the return rate is now below 13%? Use a confidence level of 95%.
Store 1 2 3 4 5 6 7 8 9 10 n x
275 250 225 200 175 150 125 100 75 50
25 22 20 18 16 13 11 9 7 4 a) State your null and alternative hypotheses. (1) Make sure I know which store you manage.
b) Test the hypothesis using a test ratio or a critical value for the observed proportion. (1) Make a diagram showing clearly where your ‘reject’ region is. (Do not round excessively. If you compute proportions carry at least 3 significant figures.) c) Find a p-value for your null hypothesis. (1) d) Test your hypothesis using an appropriate confidence interval. (2) [5] e) Using the 13% proportion as an estimate of the true proportion, find out how large a sample you need to create a 95% confidence interval with an error of no more than 1% (2)

252x0811 2/20/08 (Open in ‘Print Layout’ format) f) (Extra credit) Remember that the method that you have been using to deal with proportions substitutes the Normal distribution for the binomial distribution. In general the p-values that you have computed are lower than you would get if you used the binomial distribution. Verify this by making a continuity correction as described in the outline and repeating your test in c). (2) g) (Extra credit) Using 13%, your critical value, a point between your critical value and 13% and one or two other points on the side of the critical value implied by the alternative hypothesis (only one point on this side may give a reasonable value for a proportion) put together a power curve for your test. Remember that your standard error will change if the true proportion changes. (8) h) Go back to the test in parts a) b) and c) of this problem. Take your values of n and x and multiply them by 1.6, rounding your values to the nearest whole number (or numbers) if necessary. Find the new value of the test ratio and get a p-value. What does the change in p-value between parts c) and g) suggest about the effect of increased sample size on the power of the test? (3) [32]
Problem 4: According to Doane and Seward both the mean and the standard deviation of pH (a measure of acidity) are of interest to winemakers. Assume that your firm (store from the last problem) has gotten into the wine business. A sample of 16 wine bottles is taken. Your column has the same number as your store.
Minitab has calculated all sorts of sample statistics on your data. These are listed below. Use them.
Row C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
1 3.41 3.44 3.61 3.39 3.41 3.43 3.40 3.56 3.53 3.17
2 3.45 3.42 3.59 3.37 3.39 3.41 3.38 3.53 3.56 3.21
3 3.51 3.45 3.63 3.41 3.43 3.45 3.42 3.59 3.63 3.27
4 3.52 3.48 3.65 3.44 3.46 3.47 3.45 3.63 3.65 3.28
5 3.68 3.68 3.87 3.66 3.69 3.69 3.68 3.95 3.82 3.44
6 3.29 3.45 3.62 3.41 3.43 3.44 3.42 3.58 3.39 3.05
7 3.39 3.42 3.59 3.37 3.39 3.41 3.38 3.53 3.50 3.15
8 3.57 3.50 3.67 3.45 3.48 3.49 3.47 3.65 3.70 3.33
9 3.38 3.41 3.58 3.36 3.38 3.40 3.37 3.52 3.49 3.14
10 3.14 3.36 3.52 3.30 3.32 3.34 3.31 3.43 3.23 2.90
11 3.61 3.69 3.87 3.66 3.70 3.70 3.68 3.95 3.75 3.37
12 3.23 3.40 3.57 3.35 3.37 3.39 3.36 3.51 3.32 2.99
13 3.48 3.48 3.66 3.44 3.46 3.48 3.45 3.63 3.59 3.24
14 3.39 3.48 3.65 3.43 3.45 3.47 3.44 3.62 3.51 3.15
15 3.49 3.45 3.62 3.40 3.42 3.44 3.41 3.57 3.61 3.25
16 3.50 3.63 3.81 3.60 3.63 3.64 3.62 3.87 3.62 3.26
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
C1 16 0 3.4400 0.0347 0.1387 3.1400 3.3825 3.4650 3.5175
C2 16 0 3.4837 0.0245 0.0980 3.3600 3.4200 3.4500 3.4950
C3 16 0 3.6569 0.0259 0.1037 3.5200 3.5900 3.6250 3.6675
C4 16 0 3.4400 0.0268 0.1072 3.3000 3.3700 3.4100 3.4475
C5 16 0 3.4631 0.0281 0.1124 3.3200 3.3900 3.4300 3.4750
C6 16 0 3.4781 0.0265 0.1061 3.3400 3.4100 3.4450 3.4875
C7 16 0 3.4525 0.0278 0.1110 3.3100 3.3800 3.4200 3.4650
C8 16 0 3.6325 0.0388 0.1553 3.4300 3.5300 3.5850 3.6450
C9 16 0 3.5562 0.0382 0.1528 3.2300 3.4925 3.5750 3.6450
C10 16 0 3.2000 0.0347 0.1387 2.9000 3.1425 3.2250 3.2775
Variable Maximum
C1 3.6800
C2 3.6900
C3 3.8700
C4 3.6600
C5 3.7000
C6 3.7000
C7 3.6800
C8 3.9500
C9 3.8200
C10 3.4400

252x0811 2/20/08 (Open in ‘Print Layout’ format)
You must state H
0
and H
1
where applicable to get credit for any of the tests below. Make sure that I know which column you are using! a) The acceptable standard deviation for wine pH is 0.10. Using the data for your store, test the hypothesis that the standard deviation is 0.10 using a 95% confidence level. (2) b) Test the hypothesis that the standard deviation is below .14. (1) c) Repeat a) and b) using the sample (mean and) variance you used in a) and b) but assuming a sample size of 100. Find p-values. (4) d) Find 2-sided 95% confidence interval for the standard deviation using data from your store and assuming a sample size of 16. (2) e) Repeat d) for a sample size of 100. (1) [41] f) Here’s the easiest question on the exam. By now you should have figured out that you don’t have to understand a statistical test at all if you know i) what it assumes, ii) what the null hypothesis is and iii) what the p-value is associated with the null hypothesis. So, I am going to do a test that the standard deviation is
0.1 on the following data set.
C11
3.53 3.51 3.54 3.57 3.78 3.54 3.51 3.59 3.50 3.44 3.78
3.49 3.57 3.57 3.54 3.72
Then I am going to run a Lilliefors test on these data using Minitab. The null hypothesis of the Lilliefors test is that the sample comes from the Normal distribution. The test makes no assumptions about the mean and standard deviation of the population and computes these as sample statistics from the data. After it printed ‘Probability plot of C11,’ the computer printed a graph of the data, but the only thing I looked at was the p-value which was less than .01. After the Lilliefors test, the computer printed out the results of two versions of a statistical test on the standard deviation. The ‘Standard’ version is the method that you learned and is only applicable if the data comes from a Normal distribution. The ‘Adjusted’ version is for all other cases. So explain what p-value I look at and what it tells me.
MTB > NormTest c11;
SUBC> KSTest.
MTB > OneVariance c11;
SUBC> Test .1;
SUBC> Confidence 95.0;
SUBC> Alternative 0;
SUBC> StDeviation.
Method
Null hypothesis Sigma = 0.1
Alternative hypothesis Sigma not = 0.1
The standard method is only for the normal distribution.
The adjusted method is for any continuous distribution.
Statistics
Variable N StDev Variance
C11 16 0.100 0.0100
95% Confidence Intervals
CI for
Variable Method CI for StDev Variance
C11 Standard (0.074, 0.155) (0.0055, 0.0240)
Adjusted (0.071, 0.170) (0.0050, 0.0288)
Tests
Variable Method Chi-Square DF P-Value
C11 Standard 15.06 15.00 0.895
Adjusted 11.12 11.07 0.880