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Unformatted text preview: Engineering Statistics 1; Math 223
Test 2 (inclass portion), Summer 2004; 100 points Name Instructions:  There are problems (with parts) on this exam. The point values of each problem are listed along
with the problems.  No partial credit will be given on the true/false and multiple choice questions, #1 — 6. Each of
these problems is worth 3 points.  Since I will be giving partial credit on most problems, please show as much work as possible to
earn points. Clearly indicate your ﬁnal answers by circling them.  If you use Maple or Minitab, state this and email me your worksheets when you ﬁnish the test.
Points will be deducted, even if you supply the correct answer, if you do not have convincing
evidence to support your answer.  Number of Points Page 1 (18 points)
Page 2
(14 points) (22 points)
Page 3
(10 points) (12 points) Page 4
(12 points) Subtotal
(54 points) True/False Section. Please answer True or False in the space provided. 1. As the conﬁdence level for a conﬁdence interval increases (while all other quantities remain the same), the
width of the interval increases. 2. In analysis of variance (ANOVA), the null hypothesis is only rejected if there is a signiﬁcant difference among
all the population means. Multiple Choice Section. Please circle the best possible answer for each problem. 3. Suppose that a researcher has performed a hypothesis test and has rejected the null hypothesis at level of
signiﬁcance (1 = 0.05. Assume that no math errors have been made (that is, the observed value of the test statistic i_s in the rejection region).
(A) There’s a 5% chance that the null hypothesis is correct. (B) There’s a 5% chance that the researcher is committing an error in rejecting the null hypothesis when in fact it is
actually true. (C) There’s a 5% chance that the alternative hypothesis is false. (D) There’s a 95% chance that the researcher is making an error in rejecting the null hypothesis when in fact it is
actually true. The numbers of mosquito eggs at 20 random sites in Terre Haute are recorded. The mean number of eggs per
site can be a good indicator of how serious the mosquito season will be and how much insecticide will be needed
for spraying. The average for the last 10 years in Terre Haute is 7934.4 eggs/site/yr. This year the mean number of
eggs per site is recorded as 8675.2 eggs/site with a standard deviation of 673.9 eggs/site. Terre Haute city ofﬁcials
want to test the alternative hypothesis that this year’s mosquito situation is worse than the 10yr average. What is
the appropriate statistical test in this situation? (A) Paired ttest (B) Onesample t—test (C) Two—sample t—test (D) One sample ztest 5. If the hypothesis H0: u = 120 is rejected in favor of Ha: part 120 at at = 0.01 level of signiﬁcance, would H0 automatically be rejected at a = 0.05 level of signiﬁcance if the same sample is used? Circle the best answer. YES NO Not enough information to answer this question. 6. If the hypothesis H0: n = 120 is rejected in favor of H3: u > 120, would Ho automatically be rejected in favor of
Ha: pi 120 if the same sample is used? Assume 0t remains the same (or ﬁxed) for both tests. YES NO Not enough information to answer this question. 7. [+4 points] Newly purchased tires of a certain type are supposed to be ﬁlled to a pressure of 30 lb/inz. Suppose
we are testing H0: u = 30 versus Ha: u < 30, where it denotes the true average pressure for this type of tire. Find the
pvalue associated with the standardized test statistic value 20 = —l.75 for n = 100 sample data values. 8. [+6 points] Golf balls produced in a New Jersey plant are supposed to average 1.750 inches in diameter. A
random sample of 900 balls from the production of a single machine gives a sample mean of 1.754 inches and a
standard deviation of 0.050 inches. What is the p—value for the hypothesis test Ho: u = 1.750 versus Ha: u¢ 1.750? 9. [+4 points] A hypothesis test on one population mean was performed in Minitab. The Minitab output is shown
below. Minitab Output: OneSample: Cl Test of mu = 4 vs mu > 4 Variable N Mean StDev SE Mean Cl 5 4.600 2.074 0.927
Variable 95.0% Lower Bound T P C1 2.623 0.65 0.276 What is the rejection region for a = 0.05? That is, reject H0 if the test statistic is 10. [+6 points] A statistician claims that the average age of people who purchase lottery tickets is 70. A random
sample of 28 ticket purchasers is selected, and their ages are recorded below. The data is provided in the Minitab
worksheet that I sent to you. 49 80 24 61 79 68 63 72 46 65 76 71 90 56
70 71 71 67 82 74 39 49 69 22 56 7O 74 62 (a) Before performing any hypotheses tests, the statistician runs a normality test on the data. Is the above data
normally distributed? Why or why not? Be speciﬁc. (b) What technique did we use in class to perform inference tests on nonnormally distributed data? 11. [+4 points] We are given the following hypothesis test
Ho: u = 9.5 versus Ha: u > 9.5
and provided with the Minitab output below for analysis. Assume the data was taken from a normal distribution.
One—Sample: C1
Test ofmu = 9.5 vs mu > 9.5 Variable N Mean StDev SE Mean
C1 10 9.930 0.860 0.272 Determine the appropriate test statistic for your hypothesis test. Label it clearly as a z, t, or F test statistic. 12. [+8 points] To test the difference of physical strengths of students in physics and mathematics departments, the
number of pushups that a student could do was recorded. Twenty students from both departments were selected
randomly. Assume the distribution of the number of push—ups that mathematics (or physics) students can do is
normally distributed. The summary statistics are listed in the following table. =28.7 s=2.4
= 30.4 s = 3.8 Physics I: = 20
Mathematics n = 20 RIHI Compute an 80% conﬁdence interval for the difference between the true mean number of push—ups that
mathematics students can do and the true mean number of pushups that physics students can do. [Note: You may
assume unequal population variances for the two groups] 13. [+4 points] It is assumed when using oneway ANOVA methods that the following conditions are met:
[Please ﬁll in the missing blank] 1. Each population has a normal distribution with mean iii.
2. Each sample is randomly selected from its respective population. 3. Each sample is independent of the others 4. 14. [+6 points] On the local Joinkcom website, the following question was posed: “Do you plan on seeing
Spiderman 2 in theatres?” Out of the 93 people who responded (as of July 7, 2004 at 4 pm), 51 have or intend on
seeing Spidey in action. Based on this sample data, determine a 95% conﬁdence interval for the true proportion of
the Wabash Valley population (who are the majority of Joink subscribers) who have or will see Spiderman 2. 15. [+6 points] An efﬁciency expert wishes to determine the average time that it takes to drill holes in a certain
metal clamp. It is known from previous studies that the time required is normally distributed with standard deviation of o = 40 seconds How large a sample will she need to be 99% conﬁdent that the sample mean will be
within 5 seconds of the true mean? 16. Two different machines, A and B, used for torsion tests of steel wire were tested on 12 pairs of different types
of wire, with one member of each pair tested on each machine. The results (break angle measurements) are given below.
WireT e
MachineA 32 35 38 28 40 42 3633 37 22 42 MachineB 30 34 39 26 37 42 35 30 3O 32 20 41 The data is provided in the Minitab worksheet that I sent to you. (a) [+6 points] Set up the null and alternative hypothesis for testing if the true mean difference in the break angles
for the two machines differs from 0. (b) [+6 points] What assumption(s) will you need to make in order to perform the hypothesis test? Are all assumptions met by this data? Justify your answer by indicating the tests run and all relevant information from the
tests. (0) [+6 points] Determine the standardized test statistic for this hypothesis test. Label it clearly as a z, t, or F test
statistic. You may do this byhand or in Minitab. (d) [+4 points] Determine the approximate p—value associated with the standardized test statistic. 17. Some varieties of nematodes (round worms that live in soil and are frequently so small that they are invisible to
the naked eye) feed on the roots of lawn grasses and crops such as strawberries and tomatoes. The pest, which is
particularly troublesome in warm climates, can be treated by the application of nematocides. However, because of
the size of the worms, it is very difﬁcult to count them directly. Hence, the yield of a crop is used as a surrogate for
the number of worms. Four brands of nematocides are to be compared. Twelve plots of land of comparable fertility
that were suffering from nematodes were planted with a crop. Each nematocide was applied to three plots; the
assignment of the nematocide to the plot was made at random. At harvest time, the yields of each plot were
recorded and part of the ANOVA table appears below: Source df SS MS F~value
Nematocides  3.456  Error 8 1.200 Total 11 4. 656 (a) [+4 points] State the null and alternative hypothesis for this ANOVA test. (b) [+4 points] The value of the test statistic to test the hypothesis of no differences in the mean yields among the
four brands is: (c) [+4 points] The rejection criterion at a = 0.05 is to reject H0 if the test statistic is BONUS. [+3 points] A possible Type I error in this experiment would be to: (A) Conclude that the mean yields of the four nematocides are equal when in fact at least one is not equal.
(B) Conclude that the mean yields of the four nematocides are equal when in fact they are equal. (C) Conclude that the mean yields of the four nematocides are unequal when in fact at least one is not equal. (D) Conclude that the mean yields of the four nematocides are unequal when in fact they are equal. Engineering Statistics 1; Math 223
Test 2 (out of class portion), Summer 2004; 50 points Name Instructions: There are 14 problems on this exam. The 10 multiple choice questions are worth 3 points each (30 total
points). The point values of the other problems are listed aside the problems. Clearly indicate your ﬁnal answers by circling them. You must show your work in order to receive full
credit on any given problem. Please do your scratch work on this exam. There is no partial credit on the multiplechoice questions, so carefully check your ﬁnal answers. If you use Minitab or Maple, state this and attach your Minitab or Maple worksheets. Points will be
deducted, even if you supply the correct answer, if you do not have convincing evidence (eg., by—hand
calculations, references, Minitab worksheets, Maple worksheets) to support your answer. This exam is due by noon on Friday, July 9, 2004. You may use books, class notes, Maple, a calculator, and/or anything that is not alive [excluding myself] to
complete this exam. Do not consult each other on these problems — EVEN TO CHECK YOUR
ANSWERS! The exam is based on the honor code at RHIT. I must see your work to support your
answers. Violation of these rules will result in deduction of exam points and disciplinary action. I have not consulted with any other human being (with the exception of Dr. Evans) in completing this exam. The
answers provided on this exam are a result of my own work and effort. (Signature Required) Page 1 Number of Points J Page 4 (8 points) (12 points) Page 2 Page 5 (9 points) (5 points)
Page 6 (7 points) Subtotal (20 points) Subtotal Total (30 points) (50 points) 1. The Verizon Wireless Music Center in Indiana is planning its budget for next year. In estimating the man—hours
for security during rock concerts, the average length of rock concerts is needed. A random sample from thirty—six
other arenas was taken and the sample mean length of concerts was 160 minutes, while the sample standard
deviation was 45 minutes. A 95% conﬁdence interval for the true mean duration of music concerts is: (A) [157.94, 162.06] minutes
(B) [147.66, 172.34] minutes
(C) [71.80, 248.20] minutes (D) [145.30, 174.70] minutes (E) [157.55, 162.45] minutes 2. Suppose that a onetailed t test is being used to test H0: [,1 = 100 versus H8: [1 < 100. The level of signiﬁcance is
a = 0.05 and n = 20 observations are sampled. The rejection criterion is: (A) Reject H0 if the test statistic to is greater than 1.729 (B) Reject H0 if the test statistic to is less than 1.729 (C) Reject H0 if the test statistic to is greater than 2.039 or less than 2.039
(D) Reject Ho if the test statistic to is less than —1.725 (E) Reject Ho if the test statistic to is less than —l.645 3. A 99% conﬁdence interval for the population mean u is determined to be [65.32, 73.54] psi. If the conﬁdence
level is reduced to 90% (with all other values staying the same), the 90% conﬁdence interval for [r (A) becomes wider
(B) becomes narrower
(C) remains unchanged
(D) is empty (E) None of the above answers is correct. 4. A random sample of married people was asked “Would you remarry your spouse if you were given the
opportunity for a second time?” Of the 150 people surveyed, 127 of them said that they would do so. Find a 95%
conﬁdence interval for the true proportion of married people who would remarry their spouse. (A) [0789,0904]
(B) [0779, 0.896]
(C) [0798,0895]
(D) [0789,0887] (E) [0797,0847] 5. You have measured the systolic blood pressure of a random sample of 25 employees at your company. A 95%
conﬁdence interval for the true mean systolic blood pressure for the employees is computed to be [122, 138] mm
Hg. Which of the following statements gives a valid interpretation of this interval? (A) About 95% of the employees in the company have a systolic blood pressure between 122 and 138 mm Hg. (B) If the sampling procedure were repeated many times, then approximately 95% of the resulting conﬁdence
intervals would contain the true mean systolic blood pressure for employees in the company. (C) If the sampling procedure were repeated many times, then approximately 95% of the sample means would be
between 122 and 138 mm Hg. (D) The probability that the sample mean falls between 122 and 138 mm Hg is equal to 0.95. 6. The Indiana State Police want to estimate the average mph being traveled on the Interstate 70 over the 4th of July
weekend. If the estimate is to be within 8 mpg of the true mean with 98% conﬁdence and the estimated standard
deviation is 22 mph, how large a sample size must be taken? (A) 42 (B) 15 (C) 329 (D) 14 (E) 41 7. An experiment was conducted to compare the efﬁcacies of two drugs in the prevention of tapeworms in the
stomachs of a new breed of sheep. Five sheep were given “Drug 1” and eight sheep were given “Drug 2.” The
average worms per sheep for the ﬁve sheep given Drug 1 was 28.6, while the average worms per sheep for the eight
sheep given Drug 2 was 40.0 From previous studies, it is known that the population variances in the two groups are 0,2 = 198 (worms/sheep)2
and 022 = 232 (worms/sheep)2, respectively, and that the number of worms in the stomachs of sheep has an
approximate normal distribution. A 95% conﬁdence interval for the difference in the mean number of worms per
sheep is: (A) 11.4i 18.6 (B) 11.4:1: 18.2 (C) 41.45: 17.9 (D) 11.4 i 16.2 (E) 11.4i16.6 8. Although we have spent the majority of Engineering Statistics I talking about Type I Error (1, there is also a
Type II Error [3 in hypothesis testing. In hypothesistesting analysis, a Type 11 Error occurs only if the null
hypothesis H0 is (A) Rejected when it is true. (B) Rejected when it is false.
(C) Not rejected when it is false. (D) Not rejected when it is true. 9. Which of the following statements is/are not true? (A) If the null hypothesis is H0: u = 50, and we are given ; = 53 and cr/xr = 1.2, then the test statistic value is z =
—2.5. (B) If the alternative hypothesis has the form Ha: u > no, then an )_c value less than no certainly does not provide
support for Ha. (C) If the alternative hypothesis has the form Ha: u > in), then an ; value (such as 53.5) that exceeds no by only a
“small” amount (corresponding to 2 which is positive but small (say 2 less than 1)) does not suggest that H0 should
be rejected in favor of H3. (D) All of the above statements are true. (E) None of the above statements are true. 10. In testing the difference between the means from two normally distributed populations, H0: u] — it; = 0 versus
Ha: u, — m < 0, the summary statistics from two independent samples are: r11: 10, L = 50, 512 = 0.64, n; =10, E2 = 51, s22 = 1.86. The value of the test statistic is
(A) 2 (B) 1.61 (C) 4 (D) —1 (E) 2 11. In an experiment designed to study the effects of illumination level on task performance, nine subjects were
ﬁrst required to insert a ﬁne—tipped probe into the eyeholes often needles in rapid succession for a low light level
with black background. Then the same nine subjects did the same for a higher light level with a white background.
Each data value is the time (in seconds) required to complete the task. Sub'ect 25.01 31.05 27.47 25.74 22.96 28.84 25.85 20.89 30.05 16.61 24.98 24.59 19.68 16.07 20.84 18.23 16.50 22.96 (a) [+2 points] Set up the appropriate hypothesis test for determining whether the difference between the true
average task time under high illumination is less than the true average task time under low illumination. (b) [+2 points] What assumption(s) must you make in order to run this hypothesis test? Can you verify your
assumption(s)? If so, please do and attach your Minitab/Maple support. (c) [+2 points] Determine the value of the appropriate standardized test statistic. Is it a z or t? [You may do this
byhand or in Minitab. Please attach your Minitab output if you use Minitab] ((1) [+2 points] At level of signiﬁcance (1 = 0.05, would you reject or not reject Ho? 12. In a study of iron deﬁciency among infants, random samples of infants following different feeding programs
were compared. One group contained breastfed infants, while the children in another group were fed by a standard
baby formula without any iron supplements. Here are summary results of blood hemoglobin levels at 12 months of
age. Group Sample Size Sample Mean Sample Std. Deviation
Breast—fed 8 13 .3 1.7
Formula—fed 10 12.4 1.8 Assume that blood hemoglobin levels for each group are normally distributed and the population variances (while
unknown) cannot be assumed to be equal. ‘ (a) [+3 points] Determine a 98% conﬁdence interval for the mean difference in hemoglobin level between the two
populations of infants. (b) [+2 points] At level of signiﬁcance (1 = 0.02, can you reject H0: uBF = my in support of H3: HBF ¢ HEP? Why or
why not? 13. Light bulbs of a certain type are advertised as having an average lifetime of 800 hours. The price of these bulbs
is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be
conclusively demonstrated that the true average lifetime is smaller than what is advertised. A random sample of 50
bulbs was selected, the lifetime of each bulb determined, and the appropriate hypotheses were tested using
MINITAB, resulting in the accompanying output. Variable :1 Mean StDev SE Mean Z P
Lifetime 50 738.44 38.20 5.40 2.14 0.016 (a) [+2 points] Set up the appropriate hypotheses in this situation. (b) [+2 points] What conclusion would be appropriate for a signiﬁcance level of a = 0.05? Should the customer
buy the bulbs? 14. [+3 points] A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L
of radon. The resulting readings were as follows: 104.3 98.8 89.6 103.7 89.9 98.9
106.6 106.4 93.2 102.0 90.0 88.6 This data suggests that the population mean reading under these conditions may differ from the expected 100 pCi/L
of radon. Since the sample size is small and the data is not normally distributed, test that the population mean is not equal to
100 using bootstrapping. Report the pValue for your hypothesis test (using bootstrapping) in the space below and
attach the Minitab worksheet with your bootstrapping work. ...
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This note was uploaded on 04/24/2008 for the course MA 223 taught by Professor Devasher during the Spring '08 term at RoseHulman.
 Spring '08
 DeVasher
 Statistics

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