Download Practice Problems for Exam 1

ST 312
PRACTICE PROBLEMS FOR EXAM 1
Reiland
Topics covered on exam 1: numerical summaries, normal models, sampling distributions
for B, inference for ..
This material is covered in webassign homework assignments 1 through 4 and worksheets 1-5.
Exam information: materials allowed: calculator (no laptops or tablets), one 8 "# x 11 sheet (2 sided)
with notes, definitions, formulas, etc. Standard normal and t tables will be provided with the exam.
WARNING! The problems below may not cover all topics for which you are responsible on exam 1.
Answers are at the end of the document.
1.
The distribution of heights of American men aged 18 to 24 can be approximated by a normal model with
mean 68 inches and standard deviation 2.5 inches. Half of all young men are shorter than
a) 65.5 inches b) 68 inches c) 70.5 inches d) can't tell because the median height is not given
2.
Use the information in Problem 1 to determine the percentage of young men that are taller than 6' 1".
3.
The grade point averages (GPA) of 7 randomly chosen students in a statistics class are
3.14 2.37 2.94 3.60 1.70 4.00 1.85
Find these statistics:
a) mean
b) median and quartiles
c) range and IQR
4.
Refer to the information given in the previous problem. If (C  C)# œ 4.51, what is the standard
deviation?
5.
A standardized test designed to measure math anxiety has a mean of 100 and a standard deviation
of 10 in the population of first year college students. Which of the following observations would
you suspect is an outlier?
a) 150 b) 100 c) 90 d) 125 e) none of the above
6.
A clerk entering salary data into a company spreadsheet accidentally put an extra “0” in the boss's salary,
listing it as $2,000,000 instead of $200,000. Explain how this will affect these summary statistics for the
company payroll:
a) measures of center: mean and median
b) measures of spread: range, IQR, and standard deviation
7.
A manufacturer of television sets has found that for the sets they produce, the distribution of lengths of time
until the first repair can be approximated by a normal model with a mean of 4.5 years and a standard
deviation of 1.5 years. If the manufacturer sets the warrantee so that only 10.2% of the 1st repairs are
covered by the warrantee, how long should the warrantee last?
8.
Suppose the distribution of the amount of tar in cigarettes can be approximated by a normal model with a
mean of 3.5 mg and a standard deviation of 0.5 mg.
a. What proportion of cigarettes have a tar content that exceeds 4.25 mg?
b. In order to advertise as a low tar brand, a manufacturer must prove that their tar content is below the
25th percentile of the tar content distribution. Find the 25th percentile of the distribution of tar amounts.
9.
The mean SAT verbal score of next year's freshmen entering the local university is 600. It is also known
that 69.5% of these freshmen have scores that are less than 625. If the distribution of scores can be
approximated by a normal model, what is the standard deviation of the scores?
10. Two students are enrolled in an introductory statistics course at the University of Florida. The first student
is in a morning section and the second student is in an afternoon section. If the student in the morning
ST 312
Practice Problems Exam 1
page 2
section takes a midterm and earns a score of 76, while the student in the afternoon section takes a midterm
with a score of 72, which student has performed better compared to the rest of the students in his respective
class? Assume that both distributions of test scores can be approximated by a normal model. For the
morning class, the class mean was 64 with a standard deviation of 8. For the afternoon class, the class mean
was 60 with a standard deviation of 7.5.
11. Suppose that a normal model describes the acidity (pH) of rainwater, and the water tested after last week's
storm had a z-score of 1.8. This means that the acidity of that rain
a. had a pH of 1.8
b. varied with a standard deviation of 1.8
c. had a pH 1.8 higher than the average rainfall
d. had a pH 1.8 times that of average rainwater
e. had a pH 1.8 standard deviations higher than that of average rainwater
12. The highway gas mileage B, measured in miles per gallon (mpg), of 26 models of midsize cars, have the
following summary statistics: B œ 26.54 mpg, median œ 26 mpg, = œ 3.04 mpg, IQR œ 3 mpg. If you
convert gas mileage B from miles per gallon to B8/A which is measured in miles per liter, what are the new
values of the summary statistics? (3.785 liters œ 1 gallon).
13. Shown below is the normal probability plot for 200 monthly telephone bills.
Norm al Quantile Plot
140
phone bills
120
100
80
60
40
20
0
-4
-3
-2
-1
0
1
2
3
4
z-score
Shown below is a histogram. Is this a histogram of the same data that was used to construct the normal
probability plot?
Histogram
Frequency
40
30
20
10
0
14. A local plumber makes house calls. She charges $30 to come out to the house and $40 per hour for her
services. For example, a 4-hour service call costs $30 + $40(4) = $190.
a. The table shows summary statistics for the past month. Fill in the table to find out the cost of the service
calls.
ST 312
Practice Problems Exam 1
Statistic
Mean
Median
Stan Dev
IQR
Minimum
Hours of Service Call
4.5
3.5
1.2
2.0
0.5
page 3
Cost of Service Call
b. This past month, the time the plumber spent on a particular service call had a z-score of  1.50. What
is the z-score for the cost of the service call?
15. Which of the following is a correct statement concerning the Central Limit Theorem (CLT)?
a) The CLT states that the sample mean, x, is always equal to ..
b) The CLT states that for large samples, the sampling distribution of the sample mean is
approximately normal.
c) The CLT states that for large samples, sample mean x is equal to ..
d) The CLT states that for large samples, the sampling distribution of the population mean is
approximately normal.
e) Both c and d are correct.
16. The amount of money spent on food per week by a typical American family is known to have a mean of 92
dollars with a standard deviation 5 of 9 dollars. Suppose a random sample of 81 families is taken and the
sample mean is calculated.
a. Describe the sampling distribution model of the sample mean. (Include the mean, standard deviation,
and type of distribution if known).
b. Find the probability that the sample mean does not exceed 90.4 dollars.
17. In a learning experiment, untrained mice are placed in a maze and the time required for each mouse to exit
the maze is recorded. For untrained mice, the average time to exit the maze is . = 50 sec and the standard
deviation is 5 = 16 sec. If 64 randomly selected untrained mice are placed in the maze and the time
necessary to exit the maze recorded for each one, what is the probability that the sample mean differs from
50 by more than 3?
18. You read in a journal a report of a study that found a statistically significant result at the 5%
significance level. What can you say about the significance of this result at the 1% level?
a. it is certainly not significant at the 1% level
b. it may or may not be significant at the 1% level
c. it certainly is significant at the 1% level
19. (True or false) In a test of significance, a P-value of 0.03 means that there is only probability 0.03 that the
null hypothesis is true.
20. In a Risk Management Study on fires in compartmented fire-resistant buildings, the data below was
generated. The data in the table give the number of victims who died trying to evacuate for a sample of 14
recent fires.
ST 312
Practice Problems Exam 1
FIRE
Las Vegas Hilton (Las Vegas)
Inn on the Park (Toronto)
Westchase Hilton (Houston)
Holiday Inn (Cambridge, Ohio)
Conrad Hilton (Chicago)
Providence College (Providence)
Baptist Towers (Atlanta)
Howard Johnson (New Orleans)
Cornell University (Ithaca)
Westport Central Apartments (Kansias City, MO)
Orrington Hotel (Evanston IL)
Hartford Hospital (Hartford, CT)
Milford Plaza (New York)
MGM Grand (Las Vegas)
page 4
NUMBER OF
VICTIMS
5
5
8
10
4
8
7
5
9
4
0
16
0
36
Source: Macdonald, J. N. “Is Evacuation a Fatal Flaw in Fire-Fighting Philosophy?”
Risk Management, Vol. 33, No. 2, p. 37
a.
Construct a 98% confidence interval for the true mean number of victims per fire who die attempting to
evacuate compartmented fire-resistant buildings.
b.
Interpret the interval constructed in part a.
21. The calibration of a scale is to be checked by weighing a 10-kg test specimen #& times. The results of
different weighings are independent of one another; the sample standard deviation = of the #& weighings is
= œ !Þ#! kg. Let . denote the true average weight reading on the scale.
a. What hypotheses should be tested?
b. Suppose the rejection region is {x: x 10.11188 or x Ÿ 9.88812}. What is the value of α for this
test?
c. What is " (10.1) that is, what is T ÐX C:/ MM /<<9< A2/8 . œ "!Þ"Ñ?
d. What is the power of the test against the alternative . œ "!Þ"?
22. A T -value is the probability:
a. of committing a Type I error
b. of committing a Type II error
c. calculated assuming L! is true, that the test statistic would assume a value as or more extreme than the
value stated in the null hypothesis
d. calculated assuming L! is true, that the test statistic would assume a value as or more extreme than the
observed value of the test statistic.
e. that L! is true.
f. that we have made a mistake, assuming L! is true
23. Domino's Pizza in Big Rapids, Michigan, advertises that they deliver your pizza within 15 minutes of
placing an order or it is free. A sample of #& customers is selected at random. The average delivery time in
the sample was 13 minutes with a sample standard deviation of 4 minutes.
a. Test to determine if we can infer at the 5% significance level that the population mean is less than 15
minutes.
b Approximate the T -value for this test.
24. A marketing consultant was interested in estimating the mean weekly consumption of soft drinks among
teenagers. A random sample of '" teenagers were asked how many ounces of soft drink they consume daily.
The sum of the observations and the sum of the squared observations are shown below.
ST 312
 B3 œ 1$'& and
'"
3œ"
Practice Problems Exam 1
 ÐB3  BÑ# œ "'!&Þ$#)
page 5
'"
3œ"
Estimate with 99% confidence the mean daily consumption of soft drinks by teenagers.
25. The mean SAT mathematics score in Illinois is 450. The Chicago, Illinois school district instituted a new
high school mathematics curriculum several years ago and as a result, the superintendent thinks the Chicago
SAT math scores are above the state average. The superintendent samples 500 students and tests the
hypotheses
L! À . œ %&!
L+ À .  %&!
with α œ Þ!", where . is the mean SAT mathematics score of Chicago high school students. Historically
the standard deviation of SAT math scores has been approximately 100. If the Chicago mean SAT math
score is at least 10 points above the state average, the superintendent will receive a hefty pay raise from the
school board.
a. What is the power of this test to an increase of 10 points in Chicago's mean SAT math score?
b. What is the power of this test to an increase of 10 points in Chicago's mean SAT math score if the
sample size is 750?
26. A random sample of size 49 was selected; the resulting sample standard deviation = œ (.)#* Suppose the
confidence interval formed for . was 63 < . < 69.
a. What is the sample mean, B?
b. What is the standard error of B?
c. What was the confidence coefficient used to form the confidence interval?
27. A marketing consultant for Burpsee Cola claims that teenagers consume on average at least 22 oz. of soft
drinks per day. A random sample of '" teenagers were asked how many ounces of soft drink they consume
daily. The sample mean and sample standard deviation are, respectively, 22.38 and 5.17. Perform a
hypothesis test of the marketing consultant's claim.
ANSWERS
1. b 2. 2.28% (6' 1"=73" is how many standard deviations above the mean?)
3. a) 2.8 b) median 2.94; quartiles: since there are 7 (odd) observations, include the overall median in each half
of the data) Q" is the median of the smallest 4 observations so Q" œ "Þ)&#Þ$(
œ #Þ""à Q$ is the median of the
#
$Þ"%$Þ'
largest 4 observations so Q$ œ
œ $Þ$( c) range = 4-1.7 = 2.3; IQR = 3.37 - 2.11 = 1.26
#
4. .87 5. a
6. a) the median will probably be unaffected; the mean will be larger b) the range and standard deviation will
increase, the IQR will be unaffected
7. z = -1.27 ; x = 2.595 years. 8. a. .0668 b. z = -0.675 ; x = 3.16. 9. 0.51 = (625-600)/5 Ê 5 œ 49.02 10.
z1 =(76-64)/8=1.5 ; z2 =(72-60)/7.5=1.6 . The student in the afternoon section performed better.
11. e. 12.
B8/A œ 7.01 miles per liter; median8/A œ 6.87 miles per liter; =8/A œ .803 miles per liter; MUV8/A œ Þ793
miles per liter. 13. Yes. Note upward curvature on left portion of the normal probability plot (bills are not less
than $0); plot very “steep” in central portion, which means there are not many observations in middle of data;
downward curvature in right portion of plot indicates that the data has a shorter right tail than a normal
distribution.
14. a.
Statistic Hours of Service Call Cost of Service Call
Mean
4.5
$210
Median
3.5
$170
Stan Dev
1.2
$48
IQR
2.0
$80
Minimum
0.5
$50
b.  1.50
ST 312
Practice Problems Exam 1
15. b 16. a. the sampling distribution model is R Ð92,
*
)" Ñ
page 6
b. T ÐD Ÿ  "Þ'Ñ œ !Þ!&%)
17. T ÐB  %( or B  &$Ñ œ !Þ"$$' 18. b. 19. False
20. a. From the 14 observations, B œ 8.36, s œ 8.94; confidence coefficient .98 Ê α= .02, from the t-table,
tÞ!"ß"$ œ 2.650,
)Þ*%
x „ tαÎ#ß"$  =8  œ 8.36 „ 2.650 
 œ 8.36 „ 6.33 œ (2.03, 14.69)
"%
b. “We are 98% confident that the interval (2.03, 14.69) encloses the true mean number of victims who die
attempting to evacuate compartmented fire-resistant buildings.
21. a. H! : . œ 10, HE : . Á 10;
b. α œ P(x "!Þ""")) or x Ÿ *Þ)))"# when . œ 10) œ P(> #Þ(*'* or > Ÿ  #Þ(*'*) œ .!"
c. " (10.1) œ P(*Þ)))"# Ÿ x Ÿ "!Þ""")) when . œ 10.1) œ P(  5.297 Ÿ z Ÿ .297) œ ..6168.
d. Power when . œ "!Þ" œ "  " Ð"!Þ"Ñ œ "  Þ6168 œ Þ3832
22. d 23. a. H! À . œ "&ß L+ À .  "&à Rejection region: t < -tÞ!&ß#% = -1.711; Test statistic: > œ  #Þ&!;
Conclusion: Reject the null hypothesis. Yes, we can conclude that the mean delivery time is less than 15 minutes.
23. b. Approximate T -value: the test statistic > œ  #Þ& is between  >Þ!"ß#% œ  #Þ%*# and
 >Þ!!&ß#% œ  #Þ(*(. The T -value is between .!!& and .!" (TI83: T  @+6?/ œ .!!*))
&Þ"(
24. B „ >Þ!!&ß'! =8 = ##Þ$) „ #Þ'' 
œ ##Þ$) „ "Þ(' œ (#!.'#, #%."%)
'"
25. a. Calculate the POWER of the test when . =460 (the probability the test will reject H! when . = 460).
STEP 1: calculate the rejection region in terms of B.
Since α = .01, the RR is D  #Þ$$; in terms of B this is equivalent to the inequality
B  %&!
 #Þ$$
"!!Î&!!
which implies B  %&!  #Þ$$"!!Î&!! œ %'!Þ%#. So we will reject H! when
B  %'!Þ%#.
STEP 2: Calculate the power when . = 460.
T ÐB  %'!Þ%# A2/8 . œ %'!Ñ œ T 
B  %'!
%'!Þ%#  %'!


"!!Î&!!
"!!Î&!!
œ T D  Þ!*%
œ Þ%'$
b. If the sample size is 8 œ 750, the rejection region in terms of B is
B  %&!
 #Þ$$
"!!Î(&!
which implies B  %&!  #Þ$$"!!Î(&! œ %&)Þ&"Þ
The power when . œ 460 is
T ÐB  %&)Þ&" A2/8 . œ %'!Ñ œ T 
26. a.
'*'$
#
œ '' b. WIÐBÑ œ
=
8
B  %'!
%&)Þ&"  %'!


"!!Î(&!
"!!Î(&!
œ T D   Þ%"
œ Þ'&*
=
œ (Þ)#*
%* œ "Þ"") c. >%) 8 œ $ Ê >%) œ #Þ')$ so confidence level is
approx. 99%.
27. L! À . œ ##ß LE À .  ##, where . is the mean daily amount of soft drinks (in ounces) consumed by
teenagers.
test statistic: > œ ##Þ$)##
œ !Þ&(%;
&Þ"(
'"
P-value: in the df = 60 row, the test statistic > œ !Þ&(% is between !Þ$)(# and !Þ'()'; the area to the right of
!Þ$)(# is !Þ$&; the area to the right of !Þ'()' is !Þ#&. Therefore the lower bound on the P-value is !Þ#&,
the upper bound on the P-value is !Þ$&Þ !Þ#& Ÿ T  @+6?/ Ÿ !Þ$&.
ST 312
Practice Problems Exam 1
page 7
Conclusion: since the P-value is greater than !Þ!&, we do not reject L! à . œ ##; there is no evidence that
teenagers drink more than 22 oz. of soft drinks per day on average.