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.
1.
2.
3.
4.
5.
6.
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 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 (Make a diagram!).
b. Continuous Uniform with (Make a diagram!).
c. Binomial Distribution with .
d. Binomial Distribution with .
e. Geometric Distribution with
f. Poisson Distribution with parameter of 15.
g. Show how you would do this for a Hypergeometric Distribution with Remember .
h. (Extra credit) Hypergeometric Distribution with
i. (Extra credit) Exponential distribution with .
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) [16]
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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 Clearly state your null and alternative hypothesis for each problem. Assume that the sample is taken from a Normally distributed population.
1 16.04 257.2816
2 15.90 252.8100
3 15.81 249.9561
4 15.94 254.0836
5 15.97 255.0409
6 16.05 257.6025
7 15.91 253.1281
8 16.03 256.9609
9 15.84 250.9056
10 15.93 253.7649
11 16.04 257.2816
12 15.93 253.7649
13 15.96 254.7216
14 16.00 256.0000
15 16.16 261.1456
16 15.79 249.3241
17 15.90 252.8100
18 16.03 256.9609
19 16.03 256.9609
20 15.74 247.7476
319.00 5088.2514
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500x0712 1/8/07
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 . 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 . 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 . 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) [28]
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)[35]
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 . 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) [41]
6. (Extra credit) Repeat problem 5 using a sample of 20 of which 2 are found defective and the binomial distribution. (4)
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