MATH 227 STATISTICS FINAL EXAM - West Los Angeles College

MATH 227 STATISTICS
FINAL EXAM
M227 Spring 2007
NAME:______________
Thursday, June 7, 2007 11:30AM-1:30PM
Instructor: M. Robertson
1

M227 Fall 2006 FINAL EXAM NAME:______________
Thursday, June 7, 2007 11:30AM-1:30PM
Instructor: M. Robertson
Directions: Use the SCANTRON form for all problems 1-100. Show all of your work in the test
booklet. You may use your calculator, but indicate what you did. Good luck! You have 2 hours.
1-45) TRUE(A) or FALSE(B) 1 point each
1. For any two events A and B, PA ( ∪ B) = PA ( ) + PB ( ) −PA ( ∩ B)
.
2. If two events A and B are mutually exclusive, then PA ( ∩ B) = PA ( ) × PB ( ).
3. Zip codes are an example of numerical (not categorical) data.
2
2
4. The Chi-Squared χ test statistic is used to perform a hypothesis test on σ (or on σ ).
5. For a data set having a positively skewed unimodal histogram, the median will be to the right
of the mean.
6. In a sample of size 25, the median is the average of the 12th and 13th largest values.
2
7. The variable χ is normally distributed.
8. For any continuous probability distribution, P ( x = c)
= 0 for all values of c .
9. For a standard normal distribution, and for some point a on the number-line,
P ( z < −a)
= P(
z > a)
.
10. If the null hypothesis is not rejected, then there is strong statistical evidence that the null
hypothesis is absolutely true.
11. In the multiple regression model y = β0 + β1x1+ β2x2, β
2
can be interpreted as the amount
y will be expected to change when the value of the predictor variable x 2
is increased by one unit,
as long as the predictor variable x 1
is held constant.
2

12. The large sample z test for µ
1
− µ
2
can be used as long as at least one of the two sample
sizes, n 1 and n 2 , is greater than 30.
13. The p-value, or observed significance level, represents the probability, assuming H is false,
0
of obtaining a value of the test statistic at least as contradictory to H as what actually resulted.
0
14. If you toss a “fair” coin 100 times, you will observe exactly 50 heads.
15. If you toss a “fair” coin 100 times, the expected number of heads will be 50.
2
16. x , s and σ are sample statistics.
17. Regarding the regression equation y = β0 + β1x, the sample regression coefficient
1
ˆβ is the
unbiased estimator for population regression coefficient β
0
.
18. The standard deviation σ
x
of the sampling distribution of x increases as n increases.
19. A standard normal distribution has µ = 1 and σ = 0.
20. The mean and the standard deviation of the sampling distribution of x are µ and
respectively.
σ ,
n
21. The mean of a numerical random variable must always equal one of the possible values that
the variable can take on.
22. The 50th percentile of a normal distribution is equal to the mean of the distribution.
23. In simple linear regression, the model utility test has as its null hypothesis H0: β
1
= 0.
24. In simple linear regression, if the null hypothesis is rejected in a model utility test, there is a
useful linear relationship between x and y , so that values of x may help predict y .
3

25. In the simple linear regression model y = β0 + β1x, β
1
can be interpreted as the amount
y will be expected to change when the value of the predictor variable x is increased by one unit.
26. When a scatterplot is used to graph a bivariate data set, the variable plotted on the y-axis is
often called the response variable while the variable plotted on the x-axis is called the predictor
(explanatory) variable.
27. In Hypothesis Testing, a small p-value indicates that the observed sample results are
inconsistent with the null hypothesis.
28. The null hypothesis should be rejected when the p-value is larger than the significance level
of the test.
29. A simple linear regression model y = β0 + β1x, β
0
represents the probability of type II error
when performing a hypothesis test for β
1
.
2
30. The Chi-Squared χ test statistic is used to test independence between categorical data sets.
31. In testing the utility of a simple linear regression model, the test statistic is a t − ratio .
2
32. In categorical data analysis, a small value of the observed test statistic χ indicates that the
observed cell counts are reasonably similar to those expected when H 0 (categories are
independent) is true.
33. Categorical data used in a test for independence is often summarized in a two-way
contingency table.
34. The expected cell count for the row 1 and column 1 entry in a contingency table is equal to
the product of the row 1 and column 1 “marginal” totals.
35. Two outcomes are independent if the chance that one outcome occurs is unaffected by
knowledge of whether or not the other occurred.
36. Binomial and Poisson random variables are all examples of discrete random variables.
4

37. A discrete numerical variable is one whose possible values form an interval along the number
line.
38. The mean is the middle value of an ordered data set.
39. The Emperical Rule can only be used when the histogram of the data set can be closely
approximated by a normal curve.
40.
2
s is called the sample standard deviation.
41. A z-score tells how many standard deviations a value is from the mean.
42. If the Pearson’s correlation coefficient r between x and y is 0, then there is no relationship
between x and y.
43. A Normal curve is a skewed curve that is can be used to approximate most histograms.
44. A value of Pearson’s correlation coefficient r that is close to 1 indicates that there is a strong
linear relationship between two variables.
45. The Poisson distribution is used to determine the probability of a given number of successes
in a fixed period of time.
46-100) Multiple Choice Questions 2 points each
46. A manufacturer of cellular phones has decided that an assembly line is operating satisfactorily
if less than 3% of the phones produced per day are defective. To check the quality of a day's
production, the company decides to randomly sample 30 phones from a day's production to test
for defects. Define the population of interest to the manufacturer.
a. All the phones produced during the day in question.
b. The 30 phones sampled and tested.
c. The 30 responses: defective or not defective.
d. The 3% of the phones that are defective.
47. For a random variable z which has a standard normal distribution, P(z < 2.10) =
a) .4821 b) .0179 c) .9821 d) none of these
5

48. Suppose the random variable X has a normal distribution with mean 9.0 and variance 49.
The probability that X takes on a value of at least 18 is approximately equal to
a) .0985 b) .4015 c) .9015 d) correct approx. answer not given
49. Could the population be normally distributed
a. No, since the relationship between the expected z-values and the observed values is not linear.
b. Yes, since the relationship between the expected z-values and the observed values is
approximately linear.
c. No, since the relationship between the expected z-values and the observed values is
approximately linear.
d. Yes, since the relationship between the expected z-values and the observed values is not linear.
50. If the slope of the regression line is negative and the coefficient of determination is .64, then
the correlation coefficient is
a. .64
b. .8
c. -.64
d. -.8
6

51. Chebychev's Rule states that the proportion of observations that are within 3 standard
deviations of the mean is at least
a. 1/3
b. 2/3
c. 1/9
d. 8/9
e. correct answer not shown
52. The percentage of points falling below the 75th percentile is
a. 25%
b. 75%
c. can’t say
53. Suppose a 95% confidence interval is computed for µ resulting in the interval (112.4, 121.6).
Then it would be accurate to say
a. 95% of the time, µ falls within the interval (112.4, 121.6).
b. there is a 95% chance that µ will fall within the interval (112.4, 121.6).
c. When using this method, 95% of all the possible samples will produce intervals that do
capture µ .
d. 95% of all the possible values for µ fall within the interval (112.4, 121.6).
54. Which of the following topics did we spend the least amount of time studying in class
a. Box-and-Whisker Plots
b. Normal Distributions
c. Confidence Intervals
d. Hypothesis Testing
7

55. The diameter of ball bearings produced in a manufacturing process can be explained using a
uniform distribution over the interval 3.5 to 5.5 millimeters. What is the probability that a
randomly selected ball bearing has a diameter greater than 4 millimeters Hint: Draw the
distribution, shade appropriate area.
a. .7272
b. .4444
c. .75
d. .50
56. For air travelers, one of the biggest complaints is of the waiting time between when the
airplane taxis away from the terminal until the flight takes off. This waiting time is known to have
a skewed right distribution with a mean of 10 minutes and a standard deviation of 8 minutes.
Suppose 100 flights have randomly been sampled. Describe the sampling distribution of the mean
waiting time between when the airplane taxis away from the terminal until the flight takes off for
these 100 flights.
a. Distribution skewed right, Mean = 10 minutes, standard deviation = .8 minutes
b. Distribution skewed right, Mean = 10 minutes, standard deviation = 8 minutes
c. Distribution normal, Mean = 10 minutes, standard deviation = 8 minutes
d. Distribution normal, Mean = 10 minutes, standard deviation = .8 minutes
57. Which of the following statements about the sampling distribution of x is incorrect
a. The sampling distribution is approximately normal whenever the sample size is sufficiently
large (n > 30).
b. The sampling distribution is generated by repeatedly taking samples of size n and
computing the sample means.
c. The mean of the sampling distribution is µ
x
.
d. The standard deviation of the sampling distribution is σ
x
.
58. The average score of all pro golfers for a particular course has a mean of 70 and a standard
deviation of 3.0. Suppose 36 golfers played the course today. Find the probability that the
average score of the 36 golfers exceeded 71. Hint: This is P ( x > 71)
.
a. .1293
b. .4772
c. .3707
d. .0228
59. If we repeatedly sample from a population samples of size n, and calculate the sample
median for each sample, the accumulation of these sample medians would result in
a. a confidence interval for the population variation
b. the sampling distribution for the population mean
c. an estimate of the sample median
d. the sampling distribution of the sample median
8

60. In the construction of confidence intervals, if all other quantities are unchanged, an increase in
the sample size will lead to a _________ interval.
a. narrower
b. wider
c. less significant
d. biased
61. It is desired to estimate which of the two soft drinks, Coke or Pepsi, West Los Angeles
College students prefer. A random sample of 167 students produced the following 95%
confidence interval for the proportion of students who prefer Pepsi: (.344, .494). Identify the
point estimate for estimating the true proportion of West Los Angeles College students who
prefer Pepsi.
a. .494
b. 1.96
c. 0.95
d. .344
e. .419
62. Which of the calculated values of a test statistic would have the smallest p-value for an upper
tailed test
a. t = 3.05 with 10 degrees of freedom
b. t = 3.05 with 20 degrees of freedom
c. z = 3.05
d. z = 1.46
63. In a two-tailed large sample test with calculated test statistic z = 1.68, the p-value is
a. .0930
b. .0465
c. .9170
d. .9535
64. The probability that a normal variable X falls within 2 standard deviations of the mean is
a. .0228
b. .9772
c. .0456
d. .9544
9

65. The Central Limit Theorem predicts that
a. the sampling distribution of µ _ will be approximately normal for reasonably large samples.
x
_
b. the sampling distribution of x will be approximately normal for reasonably large samples.
_
c. the mean of the sampling distribution of x will tend to be close to µ for reasonably large
samples.
d. the mean of the sampling distribution of µ _ will tend to be close to µ for reasonably large
x
samples.
For problems 66 through 69, determine the form of the test statistic in each
hypothesis test situation. Assume the correct answer corresponds to classroom
discussion.
66. The mean GPA of West Los Angeles College football players is less than 2.3. Assume n=15
and GPAs are normally distributed.
2
a) z (standard Normal) b) t c) χ d) F e.) λ
67. The heart rate for second born identical twins are different than the heart rate for first born
identical twins. The sample consists of 90 pairs of twins.
2
a) z (standard Normal) b) t c) χ d) F e.) λ
68. The standard deviation for the volume of Odwalla soda is greater than 0.03 ounces.
2
a) z (standard Normal) b) t c) χ d) F e.) λ
69. Whether occupation is independent of religious belief, using a 2-way contingency table.
2
a) z (standard Normal) b) t c) χ d) F e.) λ
10

For problems 70 through 80, consider the following data set which consists of
measurements of the daily emission of sulfer oxides (in tons) for an industrial plant.
{ 15.8, 18.7, 6.2, 17.5, 11.0, 19.0, 26.4, 13.9, 14.7 }
The results of some of the calculations for this data set are presented below. You
may use these results to answer the following questions.
∑ x = 14320 . ∑ x 2 = 2532 . 3
70. What is n, the size of the sample
a) 9 b) 10 c) 8 d) 143.20 e) can’t say
71. What is x , the sample mean
a) 17.9 b) 15.91 c) 143.20 d) can’t say
72. x , the sample mean, and s , the sample standard deviation, are examples of
a) statistics b) parameters c) neither of these
73. What is µ , the population mean
a) 17.9 b) 15.91 c) 143.20 d) can’t say
74. x is a ________ estimator of µ .
a) biased b) interval c) unbiased d) point e) both c and d
75. What is
2
s , the sample variance
a) 253.83 b) 28.20 c) 31.73 d) 5.63 e) 5.31
76. What is s , the sample standard deviation
a) 253.83 b) 28.20 c) 31.73 d) 5.63 e) 5.31
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77. Find the 50th percentile of the above data set. What is another name for this value
a) 15.8, mode b) 15.8, median c) 20.2, range d) can’t tell
78. Considering this sample of data provided, would you describe this data set as skewed If so,
why If not, why not
a) no, since x = P50
b) yes, skewed left, since x < P50
c) yes, skewed right, since x < P50
d) yes, skewed right, since x > P50
e) none of the above
79. Which formula should be used here to find a confidence interval for the population
mean µ
x
σ
x
s
s
a) x ± z c
b) x ± zc
c) x ± t c
d) none of these
n
n
n
80. Construct the 95% confidence interval for the population mean µ
x
.
a) 15 .91±
4. 33
b) 15 .91±
3. 68
c) 15 .91±
1. 44
d) 15 .91±
4. 245
e) 15 .91±
4. 49
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81. In the formula for
2
s , the denominator is −1
n rather than n because
2
σ
x
a) dividing by n underestimates
2
b) dividing by n overestimates σ
x
c) it becomes an unbiased estimator of
d) a and c
e) b and c
2
σ
x
82. Which of the following is a measure of relative standing
a) standard deviation b) range c) z-score d) t-distribution
83. If the distribution of the random variable is _______, it is a continuous random variable.
a) binomial b) normal c) t d) a or b e) b or c
84. If X is a continuous random variable with mean µ x
= 10. 0 , is the following statement true
Explain why or why not.
P( X = 10. 0) = . 5
a) yes, the probability of the mean is always .5.
b) yes, according to the Central Limit Theorem
c) no, the probability should be 0, because the “area” of a line is 0.
⎛ n ⎞
d) no, since we don’t know n, we can’t compute ⎜ ⎟
⎝10⎠
13

For problems 85 through 90, consider the following situation - A coin is tossed 10 times.
Suppose that the coin is not fair, and that for each toss of the coin, the probability of getting
a HEAD is 0.6. Let X = the number of heads observed in the 10 tosses.
85. Is the random variable X discrete or continuous
a) discrete b) continuous
86. What possible values does X take on
a) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
b) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
c) All Real numbers in the interval [0,10]
d) .6 or .4 only
87. What type of distribution does the random variable X have
a) binomial b) uniform c) normal d) a and b e) b and c
88. Find P( X = 5 )
a) .367 b).201 c) 0 d) 1 e) 0.6
89. Find the mean of the random variable X. Show the best formula that you should use.
a) µ npq , so µ = 2. 4
x
=
b) µ
x
= np , so µ
x
= 6. 0
c) µ
x
= ∑ xp(x)
, so µ
x
= 2. 4
d) µ
x
= ∑ xp(x)
, so µ
x
= 6. 0
e) a and c are both correct
x
90. Find the standard deviation of the random variable X. Show the formula that you used.
a) σ = np( 1−
p)
= 1. 55
x
b) σ = np( 1−
p)
= 2. 4
x
∑ 2
( x − x)
c) s = = 5.5
n −1
d) none of these
_
14

For problems 91 through 94, consider the following situation -
An economist claims that, in California, the average Hispanic-owned business
generates revenues of less than $70,000 annually. A random survey of 10 Hispanicowned
business firms revealed a sample mean annual revenue of $58,600 with a
sample standard deviation of $18,000. Assume a normal population. We wish to test
the economist's claim, at a 0.10 significance level. Is there evidence that the
economist is correct
91. The alternative hypothesis that is appropriate here is
a) µ > 70000 b) µ > 70000 c) µ < 70000 d) µ = 70000
92. The distributional form of the correct test statistic to use in this situation is
a) z (standard Normal) b) t c)
2
χ
d) F
93. The decision for this hypothesis test will imply that
a) there is evidence that the economist is correct
b) there is no evidence that the economist is correct
c) correct answer not given
94. The p-value for the observed value of the test statistic is
a) greater than .1
b) between .05 and .1
c) less than .05
d) none of the choices provided are correct.
15

For problems 95 through 97, consider the following MINITAB output for analysis of a
bivariate quantitative data set. Note here that the variable Hours is being used to predict
the variable Wear.
Regression Analysis
The regression equation is
Wear = 0.0028 + 0.00657 Hours
Predictor Coef Stdev t-ratio p
Hours 0.0065723 0.0001460 45.02 0.000
R-sq = 99.1%
95. For this analysis, the coefficient of determination is
a) .991 b) .995 c) .00657 d) .0028 e) none of these
96. For this bivariate data set, the approximate value of the correlation coefficient is
a) .991 b).995 c) .00657 d) -0.991 e) none of these
97. The output indicates that Hours is a useful linear predictor of Wear. (using a .05 level of
significance.)
a) true b) false c) can't tell with the output provided
_________________________________________________________________________
98. The types of discrete random variables discussed in class were:
a) binomial
b) geometric
c) poisson
d) all of the above
e) none of these
99. Which of the following is a measure of the variability of a distribution
a. Skewness b. Median c. Standard deviation d. z-score
100. A type I error is made by
a. rejecting Ho when it is true.
b. rejecting Ho when it is false
c. failing to reject Ho when it is true
d. failing to reject Ho when it is false
16

PART III: SHOW ALL WORK FOR FULL CREDIT
95 points total
1.) Are Women Getting Taller A researcher claims that the average height of a woman
aged 20 years or older is greater than the 1994 mean height of 63.7 inches, on the basis of
data obtained from the Centers for Disease Control and Prevention’s, Advance Data
Report, No. 347. She obtains a random sample of 45 women and finds the sample mean
height to be 63.9 inches. Assume that the population standard deviation is 3.5 inches.
Test the researcher’s claim at the 0.05 level of significance.
1. Describe the population parameter about which hypothesis are to be tested.
2. The null hypothesis is H : 0
3. The alternative hypothesis is H
a
: Determine what type of test (upper, lower, or two-tailed).
4. Select the significance level α for the test. Draw the appropriate distribution, label α , and determine the
Rejection Region.
5. Display the test statistic to be used, with substitution of the hypothesized value identified in step 2 but without any
computations at this point. Also state any assumptions.
6. Compute all quantities appearing in the test statistic and then the value of the test statistic itself.
7. Determine the p-value associated with the observed value of the test statistic Draw a picture of the appropriate
distribution, shading the p-value area.
8. State the conclusion.
9. State what a TYPE II error would be in the context of this problem. For the above hypothesis test, is it
possible to find the probability of making this type of error If so, what is it
17

2.) Modern medical practice tells us not to encourage babies to become too fat. 14 females took
part in a lifelong experiment , where bivariate data was collected on each of the subjects. Let the
predictor (explanatory) variable x represent the weight (in lbs) of the subject as a 1-year old baby.
Let the response variable y represent the weight (in lbs) of the subject as a 30-year old adult. The
summary statistics are as follows:
∑ x = 300 ∑ y = 1775
xy = 38,220
∑
a.) Calculate
SS
xx
,
SS
yy
, and SS
xy
.
2
∑ x = 6572
2
∑ y = 226,125
b.) Calculate r . Interpret this value in the context of the problem.
c.) Assume the equation of the sample regression line is y = 99.25 + 1.285x. What is the
value of ˆβ 1, and interpret this value in the context of the problem.
d.) Use x = 20 in regression line to find the corresponding value for y .
e.) Give two interpretations of this corresponding y value, in terms of the above problem. Be
as specific as possible.
18

3.) Sample bivariate data indicates a correlation between the number of cigarettes a person
smokes, and the incidence of pancreatic cancer. The correlation coefficient, r, is 0.93.
Circle TRUE or FALSE to the following statements, and EXPLAIN YOUR ANSWERS.
a) There is a strong linear relationship between the number of cigarettes a person smokes and the
incidence of pancreatic cancer.
TRUE or FALSE
Explanation
____________________________________________________________________
b) The information indicates that smoking causes pancreatic cancer. TRUE or FALSE
Explanation
____________________________________________________________________
c) This regression equation relating smoking and pancreatic cancer is more useful than a
regression equation using data for which r = -0.98.
TRUE or FALSE
Explanation
____________________________________________________________________
d) The slope of the regression equation is negative. TRUE or FALSE
Explanation
_____________________________________________________________________
e) About 86% of the variation in the incidence of pancreatic cancer is due to the variation in the
number of cigarettes a person smokes.
TRUE or FALSE
Explanation
_____________________________________________________________________
f) If you were testing H0 : β
1
= 0 vs. Ha
: β1
≠ 0, you would probably reject H
0
.
TRUE or FALSE
Explanation
_____________________________________________________________________
19

4. For each of the population parameters, write the symbol for the corresponding sample
statistic (it’s point estimator), and the name of the estimator.
Estimator Symbol
Estimator Name
σ _______ ________________________
µ _______ ________________________
p _______ ________________________
2
σ _______ ________________________
µ − µ
_______ ________________________
1 2
5. IQs are calibrated so that they have a mean of 100 and a standard deviation of 15.
a) Find the probability that a randomly selected person has an IQ between 106 and
116. Be sure to draw a sketch.
Ans. _____________
b) A person with an IQ at or below the 5 th percentile is considered “developmentally
disabled”. What IQ corresponds to the cutoff for this designation
Ans. ______________
20

EXTRA CREDIT) Many people believe that criminals who plead guilty tend to get lighter
sentences than those who are convicted in trials. The data below summarizes randomly selected
sample data for San Francisco defendants in burglary cases. At the 0.05 significance level, test
the claim that the type of plea is independent of whether a person is sent to prison.
Note: First find row and column TOTALS, along with the GRAND TOTAL. To find “expected”
quantities, you may use your calculator for the calculations, but show how one of these is
calculated.
Guilty Plea Not Guilty Plea TOTAL
Sent to Prison 392 38
Not Sent to Prison 584 16
TOTAL
1. Describe the population parameter about which hypothesis are to be tested.
2. The null hypothesis is
3. The alternative hypothesis is
4. Select the significance level α for the test. Draw the appropriate distribution, label α , and determine the
Rejection Region.
5. Display the test statistic to be used, with substitution of the hypothesized value identified in step 2 but
without any computations at this point. Also state any assumptions.
6. Compute all quantities appearing in the test statistic and then the value of the test statistic itself.
7. Determine the p-value associated with the observed value of the test statistic Draw a picture of the
appropriate distribution, shading the p-value area.
8. State the conclusion.
21

EXTRA CREDIT: Sears claims that the “new and improved” batteries have mean lifetime which
is now greater than 17 months, but a leading consumer group thinks the lifetime is still 17 months
(or less).
So, the hypotheses are H 0 : µ = 17
H a ; µ > 17
a) Who is more likely to want a larger significance level, Sears or the consumer group
Explain.
Ans. ___________________________________________________________________
____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
b) Suppose both parties decide on α = 0.05. A hypothesis test is conducted from sample
data, and the p-value = 0.03. What would be the conclusion to this hypothesis test
Why Explain in the context of the problem. Your explanation should be complete and
specific.
Ans. _____________________________________________________________
____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
22