500x0712 1/8/07
SECOND EXAM
ECO500 Business Statistics
Name: _____________________
Student Number : _____________________
Remember – Neatness, or at least legibility, counts. In all questions an answer needs a calculation or
explanation to count. Show your work! Clearly state the null and alternative hypothesis in each
problem.
Part I. (12 points) Show your work! Make Diagrams!
I. (12 points) Do all the following.
x ~ N 3,11
1. P11  x  11
2. P0  x  11
3. P3  x  43 
4. Px  12 
5. P20  x  2
6. x.125
Part II. (At least 40 points. Parentheses give points on individual questions. Brackets give cumulative
point total.) Exam is normed on 50 points.
1. Find P12  x  17  for the following distributions (Use tables in c, d, f and h. All probabilities should
show 4 places to the right of the decimal point. Find the mean and standard deviation of the distribution.
(10)
a. Continuous Uniform with c  1, d  14 (Make a diagram!).
b. Continuous Uniform with c  15, d  25 (Make a diagram!).
c. Binomial Distribution with p  .45, n  25 .
d. Binomial Distribution with p  .85, n  25 .
e. Geometric Distribution with p  .15.
f. Poisson Distribution with parameter of 15.
g. Show how you would do this for a Hypergeometric Distribution with p  .45, n  25 ,
N  80. Remember M  Np .
h. (Extra credit) Hypergeometric Distribution with p  .45, n  25 , N  520 .
i. (Extra credit) Exponential distribution with c  .01 .
j. Assume that the average number of workers logging onto a system every hour is 750. What is the
chance that none will log on in a given minute?
k. What is the chance that (if the average number of workers logging onto a system every hour is
750) over 800 will logon in one hour?
2. Assume that the income in an area is Normally distributed with a known population standard deviation
of $2000. A random sample of 15 households yields a sample mean of $25000. Test the null hypothesis that
the population mean is at least $26000. Use a test ratio. Find a p-value. (5) Make 3 diagrams.
a. Show the rejection region and test the null hypothesis if the significance level is 5% (3)
b. Show the rejection region and test the null hypothesis if the significance level is 1% (1)
c. Find a p-value for the null hypothesis and compare the p-value to the results of a) and b). Make a
diagram. (2)
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500x0712 1/8/07
3. The claimed mean weight of a can in a batch
of canned vegetables is 16 oz. You take a sample
of 20 cans and find the weights at right.
There are 9 parts to this problem. Note that the
absolute value of the test ratio I got in part a) was
2.174, you should be very close. Please do not
round excessively or your answers will be way
off. Use   .05 . Clearly state your null and
alternative hypothesis for each problem. Assume
that the sample is taken from a Normally
distributed population.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
x2
16.04 257.2816
15.90 252.8100
15.81 249.9561
15.94 254.0836
15.97 255.0409
16.05 257.6025
15.91 253.1281
16.03 256.9609
15.84 250.9056
15.93 253.7649
16.04 257.2816
15.93 253.7649
15.96 254.7216
16.00 256.0000
16.16 261.1456
15.79 249.3241
15.90 252.8100
16.03 256.9609
16.03 256.9609
15.74 247.7476
319.00 5088.2514
a) Test the hypothesis that the mean is less than 16 using a test ratio. Make a diagram showing your
rejection regions. Find an approximate p-value for the test ratio. (2)
b) Test the hypothesis that the mean is less than 16 using a critical value for x . Make a diagram showing
your rejection regions. (2)
c) Test the hypothesis that the mean is less than 16 using a confidence interval for the population mean.
Make a diagram showing your confidence interval. (2)
d) Test the hypothesis that the mean is equal to 16 using a test ratio. Make a diagram showing your rejection
regions. Find an approximate p-value for the test ratio. (1)
e) Test the hypothesis that the mean is equal to 16 using a critical value for x . Make a diagram showing
your rejection regions. (1)
f) Test the hypothesis that the mean is equal to 16 using a confidence interval for the population mean.
Make a diagram showing your confidence interval. (1)
g) Test the hypothesis that the mean is greater than 16 using a test ratio. Make a diagram showing your
rejection regions. Find an approximate p-value for the test ratio. (1)
h) Test the hypothesis that the mean is greater than 16 using a critical value for x . Make a diagram showing
your rejection regions. (1)
i) Test the hypothesis that the mean is greater than 16 using a confidence interval for the population mean.
Make a diagram showing your confidence interval. (1)
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4. a) Use a test ratio and the sample standard deviation you computed in problem 3 to test if the population
standard deviation is 0.15 (3)
b) Confirm your results by creating a confidence interval for the variance. (1)
c) Repeat the test assuming that the sample size is 40. (2)
d) Confirm your results by creating a confidence interval for the variance. (1)
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5. If out of a sample of 100 parts, ten are found defective, test the hypothesis that the proportion of
defective parts in the population is no more than 5%.
a) Test the hypothesis that using a test ratio. Make a diagram showing your rejection regions. Find an
approximate p-value for the test ratio. (2)
b) Test the hypothesis using a critical value for p . Make a diagram showing your rejection regions. (2)
c) Test the hypothesis using a confidence interval for the population proportion. Make a diagram showing
your confidence interval. (2)
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6. (Extra credit) Repeat problem 5 using a sample of 20 of which 2 are found defective and the binomial
distribution. (4)
2