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Study Sheet and Sample Problems for Test 3 by jasonpeters

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```									Math 308 Spring 2009
Study Sheet and Sample Problems for Test 3

Text for TEST 3           Assigned                Other material
7.1 Point estimation      5
7.2 Interval              17
estimation
Hypothesis testing: Understand the hypothesis-testing five-step
process. Know the hypotheses for tests about one mean, two
means and more than two means. Know which test statistics to
use based on the assumptions of the test statistic. Know how to
relate the alternate hypothesis to the decision rule (rejection
region). Know how to calculate the test statistic from a formula
sheet.
7.3 Tests of              Assigned                Class Handout:
Hypotheses                                        Statistical Testing
7.4 Null hypotheses       7.30, 7.31, 7.35,       Class Handout:
and tests of              7.37,7.38               Statistical Testing
hypotheses
7.5 Hypotheses about 7.39, 7.43, 7.48             Class Handout:
one mean                                          Statistical Testing
7.8 Inference             7.68, 72                Class Handout:
concerning the                                    Statistical Testing
difference between
two means
9.1-9.3                   4, 11, 22, 27
12.1,12.2 ANOVA:          Example on              Online class notes:
One-way ANOVA             Handout                 Analysis of
(Completely                                       Variance
randomized design)
11.1-11. Simple           Example on              Online class notes:
Linear Regression         Handout                 Regression
Old Test Questions
1. A random sample of the number of MotoCars made at the Electric MotoCar
Factory yielded the numbers in the table. The graph is a histogram of the number
of MotoCars made for the twenty randomly chosen days. The mean daily
number of cars was 36.75 and the standard deviation was 6.71115. The owners of
Electric MotoCar want to know whether the actual (population) mean is greater
than 34 cars/day. Fill in the five steps of the hypothesis-testing process for testing
this hypothesis. Use a significance level of 0.05.
MotoCars
8
36.00
28.00
49.00
44.00                      6
35.00
37.00
Frequency

36.00
35.00                      4
37.00
36.00
33.00
39.00                      2

40.00
32.00                                                           Mean = 36.75
31.00                                                           Std. Dev. = 6.71115
N = 20
41.00                      0
20.00   30.00        40.00   50.00
42.00                                      MotoCars
30.00
23.00
51.00
(a) Hypotheses[3 points]:
(b) Significance level and test statistic (give your reasons for choosing the test
statistic) [7]:
(Question 1 continued)
(c) Decision rule (rejection region) 4]:
(d) Computed sample statistic [8]:
(e) Decision (include reason for decision)[2]:
2. [8] Refer to problem 1. What is the probability of a Type II error if   35.45
MotoCars daily?
3. [10] Refer again to problem 1. Give the 95% confidence interval for the actual
number of MotoCars produced daily.
4. [5] If we want to determine the average mechanical aptitude of a large group of
workers, how large a random sample will we need to be able to assert with
probability 0.95 that the sample mean will not differ from the true mean by more
than 4 points? Assume that it is known from past experience that   25.0
5. [25] The number of cars manufactured at Factory A was collected for 16
randomly selected days. The number of cars manufactured by Factory B was
collected for 19 randomly selected days. The table presents sample statistics for
the two random samples from the two competing factories. The histograms
present the sample frequency distribution of Factory A and Factory B.
Factory A      Factory B
Mean = 206 Mean = 209
Std Dev        Std Dev
=1.21106         =2.000
n=16           n=19

Histogram                                                                         Histogram

for Factory= A                                                                     for Factory= B
6                                                                                  5

5
4

4
Frequency
Frequency

3

3

2
2

1
1

Mean = 206.00                                                                      Mean = 209.00
Std. Dev. = 1.21106                                                                Std. Dev. = 2.00
0                                                N = 16                            0                                                N = 19
204.00   205.00   206.00   207.00   208.00                                         206.00   208.00          210.00     212.00
Cars                                                                               Cars

Perform a complete five-step hypothesis-testing procedure (including your
conclusion) for the null hypothesis H 0 :  A   B . Let   .05 . Number your
steps. Explain your answers where appropriate. (Use problem 1 as a reference.)
6. [28] Given the following observations from four independent random samples,
complete the five-step hypothesis-testing procedure for comparing the four means
corresponding to the four Treatments. Number your steps and use problem 1 as a
general reference for an appropriate answer.
Treatment 1         Treatment 2        Treatment 3        Treatment 4
6                  13                  7                 3
4                  10                  9                 6
5                  13                 11                 1
12                                    4
1
Source        Sum of     Degrees of     Mean Square       F
Squares freedom
Treatment
Error

OTHER SAMPLE PROBLEMS FOR TEST 3: 1. The following figures are the
number of widgets filled per day at two competing plants.

Plant A          Plant B
19     23       17      19
21     21       15      15
15     17       12      11
17     12       12      10
24     16       16      20
12     15       15      12
19     20       11      13
14     18       13      17
20     14       14      16
18     22       21      18
Mean = 17.85    Mean = 14.85
Std Dev = 3.50 Std Dev = 3.15

With the information you have been given so far, how would you verify that each sample
is from a normal population.
How would you find the probability of a sample mean of 17.85 or larger for Plant A if
you assume that the actual mean for Plant A is 15?
2. A cola dispensing machine at XYZ company dispenses 9 ounce cups of cola. An
employee of XYZ thinks that the machine is cheating consumers. A random sample of 50
cups has a mean of 8.7 ounces and a standard deviation of 0.5 ounces. If the mean
volume of cola dispensed is actually 9 ounces, what is the probability of a mean this
small or smaller?

5.       The functional reach of adult men is a normally distributed random
variable with mean 32.33 inches and standard deviation 1.63 inches.
(a)         What is the probability that the reach of a randomly selected man
exceeds 34.5 inches?
(b)         A second man is randomly selected. What is the probability that both
men have reaches exceeding 34.5 inches? (Assume independence.)
(c)         What is the probability that the reach of a randomly selected man is
between 31.515 inches and 33.96 inches?
6. The Leakey Brothers offer a one-year warranty on radiator replacements. The
time to failure of a Leakey radiator has an exponential distribution with mean
process rate λ= 0.6 per year.
(a) What is the probability of a radiator failing within the warranty period?
(b) If the Leakey Brothers make a profit of \$125 on every radiator that survives
the warranty period and lose \$50 on each radiator that fails under warranty, what
is the expected profit per radiator sold?

Some sample questions for Test 3: Note that there is no ANOVA question.

1. The following figures are the number of widgets produced per day at two competing plants.

Plant A             Plant B
19        23        17        19
21        21        15        15
15        17        12        11
17        12        12        10
24        16        16        20
12        15        15        12
19        20        11        13
14        18        13        17
20        14        14        16
18        22        21        18
Mean = 17.85        Mean = 14.85
Std Dev = 3.50      Std Dev = 3.15

A. With the information you have been given so far, how would you verify that each sample is from a
normal population. Why is this “check” important?

B. Find the 95% CI for the mean of the number of widgets made daily by Plant A.
C. Test the hypothesis that Plant A makes more widgets daily than Plant B.

2. A cola dispensing machine at XYZ company dispenses 9 ounce cups of cola. An employee of XYZ
thinks that the machine is cheating consumers. A random sample of 50 cups has a mean of 8.7 ounces and a
standard deviation of 0.5 ounces.

A. Construct a 95% confidence interval for the average amount of cola dispensed by this machine.

B. Test the hypothesis that the actual mean volume of cola is less than 9 oz.

3. How would a 95% confidence interval compare to the 99% interval?

a) wider
b) same width
c) narrower

4. The production foreman of Blue Ox Brewing Company is worried that the bottling process is overfilling.
Each bottle of Blue Ox Beer is supposed to contain 12 oz of fluid. A random sample of 40 bottles is taken
from bottles coming off the production line, and the contents of each bottle are carefully measured. It is
found that the mean amount of beer for the sample of bottles is 12.1 oz with a standard deviation of 0.2 oz.

A. What is the 99% CI for μ?
B. Test the hypothesis that the mean amount of beer is greater than 12 oz.

5. Plant therapists believe that plants can reduce the stress levels of humans. At Christian Brothers
University, a random sample of 10 undergraduates took part in a study to determine what effect the
presence of a plant has on relaxation. Each student participated in two sessions – one with a plant and one
without a plant. During each session, finger temperature was measured (increasing finger temperature
indicates an increased level of relaxation). You are asked to analyze the data. Some of your results are
presented here.

d = Plant – NoPlant for an individual
d = mean difference in finger temperature
95.0 % C.I. for d : (-1.975, 1.995)
A. Based on the confidence interval results, can the researchers state that plants help college students to
relax? Explain.
B. You are asked to test the hypothesis that the presence of a plant increases relaxation.
Explain how you would do the test. Give all five steps.

6. The caffeine counts (in mg) are given below for a dozen cans of King of Caffeine cola randomly selected
from the production line. The sample has a mean of 33.05 and a standard deviation of 1.13.
34.2   33.7    31.9       34.3   31.6   32.7
33.1   35.2    31.6       32.9   33.0   32.4

Test the hypothesis that the mean is greater than 32 mg.

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