1. True or False. Justify for full credit.
a. If the variance from a data set is zero, then all the observations in this data set must be identical.
(b) P(A AND Ac ) 1, where Ac is the complement of A.
c. The mean is always equal to the median for a normal distribution.
d. A 99% confidence interval is wider than a 95% confidence interval of the same parameter.
e. It is easier to reject the null hypothesis if we use a smaller significance level α.
Refer to the following frequency distribution for Questions 2, 3, 4, and 5. Show all work. Just the answer, without supporting work, will receive no credit.
A random sample of 25 customers was chosen in UMUC MiniMart between 3:00 and 4:00 PM on a Friday afternoon. The frequency distribution below shows the distribution for checkout time (in minutes).
Checkout Time (in minutes)
Frequency
Relative Frequency
1.0 - 1.9
4
2.0 - 2.9
0.4
3.0 - 3.9
4.0 - 4.9
5
Total
25
2.Complete the frequency table with frequency and relative frequency.
3.What percentage of the checkout times was at least 3 minutes?
4.In what class interval must the median lie? Explain your answer.
5. Assume that the largest observation in this dataset is 4.8. Suppose this observation were incorrectly recorded as 8.4 instead of 4.8. Will the mean increase, decrease, or remain the same? Will the median increase, decrease or remain the same? Why?
Refer to the following information for Questions 6, 7, and 8. Show all work. Just the answer, without supporting work, will receive no credit.
A 6-faced die is rolled two times. Let A be the event that the outcome of the first roll is an even number, and B be the event that the outcome of second roll is greater than 4.
6. How many outcomes are there in the sample space?
7. What is the probability that the outcome of the second roll is greater than 4, given that the first roll is an even number?
STAT 200: Introduction to Statistics Final Examination, Spring 2015 OL4/US2
Page 3 of 6
8.
Are A and B independent? Why or why not?
Refer to the following situation for Questions 9, 10, and 11.
The five-number summary below shows the grade distribution of two STAT 200 quizzes.
Minimum
Q1
Median
Q3
Maximum
Quiz 1
12
40
60
95
100
Quiz 2
20
35
50
90
100
For each question, give your answer as one of the following: (a) Quiz 1; (b) Quiz 2; (c) Both quizzes have the same value requested; (d) It is impossible to tell using only the given information. Then explain your answer in each case. (5 pts each)
9. Which quiz has less interquartile range in grade distribution?
10. Which quiz has the greater percentage of students with grades 90 and over?
11. Which quiz has a greater percentage of students with grades less than 60?
Refer to the following information for Questions 12 and 13. Show all work. Just the answer, without supporting work, will receive no credit.
There are 1000 juniors in a college. Among the 1000 juniors, 200 students are in the STAT200 roster, and 100 students are in the PSYC300 roster. There are 80 students taking both courses.
12. What is the probability that a randomly selected junior is taking at least one of these two courses? (10 pts)
13. What is the probability that a randomly selected junior is taking PSYC300, given that he/she is taking STAT200? (10 pts)
14. UMUC Stat Club is selecting three officers for the school year - a president, a vice president and a treasurer. There are 10 qualified candidates. How many different ways can the officers be selected? Show all work. Just the answer, without supporting work, will receive no credit.
15. Imagine you are in a game show. There are 6 prizes hidden on a game board with 10 spaces. One prize is worth $100, another is worth $20, and four are worth $5. You have to pay $20 to the host if your choice is not correct. Let the random variable x be the winning. Show all work. Just the answer, without supporting work, will receive no credit.
STAT 200: Introduction to Statistics Final Examination, Spring 2015 OL4/US2
Page 4 of 6
(a) What is your expected winning in this game?
(b) Determine the standard deviation of x. (Round the answer to two decimal places)
16. Mimi just started her tennis class three weeks ago. On average, she is able to return 30% of her opponent's serves. Assume her opponent serves 10 times. Show all work. Just the answer, without supporting work, will receive no credit.
a. Let X be the number of returns that Mimi gets. As we know, the distribution of X is a binomial
probability distribution. What is the number of trials (n), probability of successes (p) and
probability of failures (q), respectively?
(b) Find the probability that that she returns at least 1 of the 10 serves from her opponent.
(c) How many serves can she expect to return?
Refer to the following information for Questions 17, 18, and 19. Show all work. Just the answer, without supporting work, will receive no credit.
The heights of pecan trees are normally distributed with a mean of 10 feet and a standard deviation of 2 feet.
17. What is the probability that a randomly selected pecan tree is between 10 and 12 feet tall?
18. Find the 3rd quartile of the pecan tree height distribution.
19. If a random sample of 100 pecan trees is selected, what is the standard deviation of the samplemean?
20. A random sample of 225 SAT scores has a sample mean of 1500. Assume that SAT scores have a population standard deviation of 300. Construct a 95% confidence interval estimate of the mean SAT scores. Show all work. Just the answer, without supporting work, will receive
no credit.
(10 pts)
21.
Consider the hypothesis test given by
H 0 : p
0.5
H1 : p
0.5
ˆ
0.53 .
In a random sample of 225 subjects, the sample proportion is found to be p
a. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit.
b. Determine the p-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit.
(c) Is there sufficient evidence to justify the rejection of H0 at the 0.01 level?Explain.
22. Consumption of a large amount of alcohol is known to increase reaction time. To investigate the effects of small amounts of alcohol, reaction time was recorded for five individuals before and after 2 ounces of alcohol was consumed by each. Does the data below suggest that the consumption of 2 ounces of alcohol increases mean reaction time?
Reaction Time (seconds)
Subject
Before
After
1
6
7
2
8
8
3
4
6
4
7
10
5
9
10
Assume we want to use a 0.1 significance level to test the claim.
a. Identify the null hypothesis and the alternative hypothesis.
b. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit.
c. Determine the p-value. Show all work; writing the correct critical value, without supporting work, will receive no credit.
d. Is there sufficient evidence to support the claim that the consumption of 2 ounces of alcohol increases mean reaction time? Justify your conclusion.
23. A STAT 200 instructor is interested in whether there is any variation in the final exam grades between her two classes Data collected from the two classes are as follows:
Her null hypothesis and alternative hypothesis are:
a. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit.
b. Determine the p-value for this test. Show all work; writing the correct P-value, without supporting work, will receive no credit.
(c) Is there sufficient evidence to justify the rejection of H0 at the
0.01 level?
Explain.
STAT 200: Introduction to Statistics
Final Examination, Spring 2015 OL4/US2
Page 6 of 6
24.
A random sample of 4 professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of
dollars).
x
0
1
3
5
y
1
2
3
8
a. Find an equation of the least squares regression line. Show all work; writing the correct equation, without supporting work, will receive no credit.
b. Based on the equation from part (a), what is the predicted value of y if x = 4? Show all work and justify your answer.
25. The UMUC Daily News reported that the color distribution for plain M&M's was: 40% brown, 20% yellow, 20% orange, 10% green, and 10% tan. Each piece of candy in a random sample of 100 plain M&M's was classified according to color, and the results are listed below. Use a 0.05 significance level to test the claim that the published color distribution is correct. Show all work and justify your answer.
Color
Brown
Yellow
Orange
Green
Tan
Number
42
21
12
7
18
a. Identify the null hypothesis and the alternative hypothesis.
b. Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive no credit.
c. Determine the p-value. Show all work; writing the correct critical value, without supporting work, will receive no credit.
d. Is there sufficient evidence to support the claim that the published color distribution is correct? Justify your answer.
STAT 200 QUIZ 3
There are 23 problems. Mark only ONE choice in each problem.
01: An assumption made about the value of a population parameter is called a
A. conclusion
B. confidence
C. significance
D. hypothesis
E. none of the above
02: The p-value is a probability that measures the support or lack of support for the
A. null hypothesis
B. alternative hypothesis
C. either the null or the alternative hypothesis
D. sample statistic
E. none of the above
03: In a t-test, which of the following does not need to be known in order to compute the p-value?
A. knowing whether the test is one-tail or two-tails
B. the value of the test statistic
C. the level of significance
D. the Degrees of Freedom
E. none of the above
04: As the test statistic becomes larger, the p-value
A. becomes smaller
B. becomes larger
C. becomes negative
D. stays the same because the sample size has not been changed
E. none of the above
05: What type of error occurs if you fail to reject H0 when, in fact, it is not true?
A. Type I
B. Type II
C. Either Type I or Type II, depending on the level of significance.
D. Either Type I or Type II, depending on whether the test is one tailed or two tailed.
E. None of the above.
06: The level of significance α is the
A. same as the p-value
B. 1 - β
C. maximum allowable probability of Type II error
D. maximum allowable probability of Type I error
E. none of the above
07: In hypotheses testing if the null hypothesis has been rejected when the alternative hypothesis is true, then
A. a Type I error has been committed
B. a Type II error has been committed
C. either a Type I or Type II error has been committed
D. the correct decision has been made
E. none of the above
08: In a one-tail, left-tail z-test, the test statistic is z = -2.01. The p-value is
A. 0.0179
B. 0.0228
C. 0.0222
D. 0.0174
E. none of the above
09: In a two-tail z-test, the test statistic is z = -2.5. The p-value for this test is
A. -1.25
B. 0.4938
C. 0.0062
D. 0.0124
E. none of the above
10: In a two-tail t-test, the test statistic t = -2.89 and the sample size is 30. The p-value is
A. 0.9928
B. 0.0019
C. 0.0038
D. 0.0036
E. 0.0072
11. In one-tail, left-tail t-test, the test statistic t = -1.58 and the sample size is 23. The p-value is
A. 0.0571
B. 0.9358
C. 0.0642
D. 0.1284
E. 0.0639
12: The average life expectancy of a certain brand of tire has been 40,000 miles. Because of a new manufacturing process introduced it is believed that the life expectancy has increased. To test the validity of this the correct set of hypotheses is
A. H0: µ < 40,000 HA: µ ≥ 40,000
B. H0: µ = 40,000 HA: µ ≥ 40,000
C. H0: µ > 40,000 HA: µ ≤ 40,000
D. H0: µ = 40,000 HA: µ < 40,000
E. H0: µ = 40,000 HA: µ > 40,000
13: What kind of Alternative Hypothesis is used in the figure below?
A. HA: µ = µ0
B. HA: µ ≠ µ0
C. HA: µ > µ0
D. HA: µ < µ0
E. none of the above
14. Which type of Alternative Hypothesis is used in the figure below?
A. HA: µ = µ0
B. HA: µ ≠ µ0
C. HA: µ > µ0
D. HA: µ < µ0
E. none of the above
15: A soft drink filling machine, when in perfect adjustment, fills the bottles with 12 ounces of soft drink. If the machine overfills or under fills then it must be shut down and readjusted. To determine whether or not the machine is properly adjusted, the correct set of hypotheses is
A. H0: µ < 12 HA: µ ≥ 12
B. H0: µ = 12 HA: µ > 12
C. H0: µ ≠ 12 HA: µ = 12
D. H0: µ = 12 HA: µ ≠ 12
E. H0: µ = 12 HA: µ < 12
Use the following information to do the next four problems.
A random sample of 100 people was taken. In the sample 85% favored Candidate Green. We are interested in determining whether or not the proportion of the population in favor of Green is significantly more than 80%.
16: The correct set of hypotheses for this problem is
A. H0: p = 0.85 and HA: p > 0.85
B. H0: p > 0.80 and HA: p = 0.80
C. H0: p = 0.80 and HA: p > 0.80
D. H0: p = 0.80 and HA: p ≥ 0.80
E. H0: p ≠ 0.80 and HA: p > 0.80
17: Find the test statistic.
A. 0.80
B. 0.05
C. 1.25
D. 2.00
E. none of the above
18: Find the p-value.
A. 0.1071
B. 0.8929
C. 0.8944
D. 0.1056
E. none of the above
19: At the 5% level of significance, can we conclude that the proportion of the population in favor of Green
A. is significantly greater than 80%
B. is not significantly greater than 80%
C. is significantly greater than 85%
D. is not significantly greater than 85%
E. none of the above
Use the following information to do the next four problems.
The sales of a grocery store had an average of $8,000 per day. The store started an aggressive advertizing campaign in order to increase sales. To determine whether or not the advertizing campaign has been effective, i.e. the sales increased, a sample of 66 days was selected. For this sample the mean was $8,300 per day and the standard deviation $1,200.
20: The correct Alternative Hypothesis for this problem is
A. HA: x‾ > 8,000
B. HA: x‾ ≥ 8,000
C. HA: µ > 8,300
D. HA: µ ≥ 8,000
E. HA: none of the above
21: The value of the test statistic is
A. -2.03
B. 2.03
C. -2.00
D. 2.00
E. none of the above
22: The p-value is
A. 0.0248
B. 0.0228
C. 0.9752
D. 0.9768
E. 0.0232
23: At the 5% level of significance, can we conclude that the advertizing campaign
A. increased sales?
B. decreased sales?
C. did not much affect sales?
D. was not worth the cost?
E. none of the above
STAT 200 QUIZ 6
01. A probability is a number p such that:
A. 0 < p < 1
B. 0 ≤ p ≤ 1
C. -1 < p < 1
D. -1 ≤ p ≤ 1
E. None of the above
02. The Complement rule states that the probability of an event not occurring is
A. equal to one minus the probability it will occur.
B. equal to one minus the probability it will not occur.
C. equal to 0.0
D. equal to 1.0
E. None of the above
03. A tire manufacturer claims that the probability of its XLT tire lasting 50,000 miles or more is 0.80. If three XLT tires are installed on a car what is the probability that all three will last 50,000 miles or more?
A. 0.800.
B. 0.640.
C. 0.240.
D. 0.512.
E. None of the above
04. In how many different ways can a work party of 4 be chosen from 9 volunteers?
A. 4
B. 36
C. 126
D. 3024
E. None of the above
05. Which of the following is/are properties of the Binomial Distribution?
A. The number of trials is fixed in advance.
B. The trials are independent.
C. Each trial has exactly two outcomes.
D. The probability of success is the same for each trial.
E. All of the above.
06. A fair coin is tossed 6 times. What is the probability of getting exactly 3 heads in the 6 tosses?
A. 0.2
B. 0.3
C. 0.4
D. 0.5
E. None of the above
07. The mean of a probability distribution is referred to as the
A. median
B. mode
C. expected value
D. weighted mean
E. none of the above
08. The number of armed robberies and their probabilities, in a particular city in a month, are given in the table below:
Number of armed robberies
Probability
1
0.05
2
0.30
3
0.40
4
0.25
How many armed robberies on the average should be expected on a typical month?
A. 10.00
B. 2.50
C. 2.85
D. 3.01
E. None of the above
09. A recent survey of local cell phone retailers showed that of all cell phones sold last month, 64% had a camera, 28% had a music player and 22% had both. The probability that a cell phone sold last month had a camera or a music player is
A. 0.92
B. 0.70
C. 0.18
D. 0.36
E. None of the above
10. The probability of two events occurring together is referred to as
A. a marginal probability
B. a conditional probability
C. a subjective probability
D. the multiplication rule
E. a joint probability
11. A student's score on a test is 110. The scores are normally distributed with mean µ = 120 and standard deviation σ = 8. Find the student's z-score.
A. 1.25
B. -1.25
C. -1.52
D. 0.25
E. None of the above
12. To construct a normal distribution, the measurements needed are
A. the mean and the median
B. the mode and the standard deviation
C. the median and the standard deviation
D. the standard deviation and the variance
E. none of the above
13. Which of the following is not a property of the standard normal distribution?
A. It is continuous
B. It is uniform
C. It is bell-shaped
D. It is unimodal
E. The curve never touches the horizontal axis.
14. Consider the Standard Normal distribution. Find P(-0.73 < z < 2.21)
A. -0.7537
B. 0.9987
C. 0.7534
D. 0.7537
E. none of the above
15. Consider the Standard Normal distribution. Find the probability P(z > 0.59)
A 0.7224
B 0.2190
C 0.2224
D 0.2776
E None of the above
16. Consider the Standard Normal distribution. Find the probability that z is less than -1.82.
A. 0.0351
B. -0.0344
C. 0.0344
C.0.9656
E. none of the above
17. Find the value of z such that the area to the left of z is 9% of the total area under the curve.
A. 0.82
B. -0.82
C. 1.34
D. -1.34
E. none of the above
18. Find the value of z such that the area to the right of z is 67% of the total area under the standard normal curve.
A 0.44
B -0.44
C 1.00
D -1.50
E None of the above
19. Find the value of z such that the area between -z and +z is 98% of the total area under the standard normal curve.
A. 2.02
B. 1.96
C.
Correct answer 2.33 (to be precise 2.326)
20. The mean amount spent by a family of four on food per month is $500 with a standard deviation of $75. Assuming a normal distribution, what is the probability that a family spends more than $410 per month?
A. 0.1151
B. 0.1539
C. 0.8849
D. 0.8461
E. none of the above
21. If a population is normally distributed, the distribution of the sample means for a given sample size n will
A. be positively skewed.
B. be negatively skewed.
C. be uniform.
D. be normal.
E. none of the above
22. If a population is not normally distributed, the distribution of the sample means for a given sample size n will
A. take the same shape as the population.
B. approach a normal distribution as n increases.
C. be positively skewed.
D. be negatively skewed.
E. none of the above
23. An auditor takes a random sample of size n=110 from a large population. The population standard deviation is not known, but the sample standard deviation is s = 48. Find the standard error of the mean.
A. 48
B. 4.58
C. 0.44
D. 2.29
E. None of the above
24. The scores on a certain test are normally distributed with mean 61 and standard deviation 3. What is the probability that a sample of 100 students will have a mean score more than 61.3?
A. 0.4602
B. 0.3413
C. 0.8413
D. 0.1587
E. none of the above
25. A random sample of 66 observations was taken from a large population. The population proportion is 12%. The probability that the sample proportion will be more than 17% is
A. 0.0568
B. 0.8944
C. 0.1056
D. 0.4222
E. none of the above
26. When the 99% confidence interval is calculated instead of the 95% confidence interval with the sample size n being the same the margin of error will be
A. smaller.
B. larger.
C. the same.
D. reduced by 4%
E. none of the above
27. As the sample size increases,
A. the population standard deviation decreases
B. the population standard deviation increases
C. the standard error increases.
D. the standard error decreases.
E. none of the above
28. Statistical inference
A. is reasoning from a sample to a population.
B. is reasoning from a population to a sample.
C. requires a large sample.
D. requires examination of the entire population.
E. is based on deductive reasoning.
29. The best point estimate of the population mean is
A. the sample mean
B. the sample median
C. the sample mode
D. the sample midrange
E. none of the above.
30. A 95% confidence interval based on a sample for the mean time it takes to process a claim by an insurance company is between 10 and 15 days. This means that
A. only 5% of all claims take less than 10 or more than 15 days to process.
B. only 5% of all claims take between 10 and 15 days to process.
C. about 95% of all intervals similarly constructed from samples of the same size will contain the true population mean processing time.
D. the probability is 0.95 that all claims take between 10 and 15 days o process.
E. none of the above
31. A sample of 31 people was randomly selected from among the workers in a large shoe factory. The time taken for each person to polish a finished shoe was measured. The sample mean was 2.2 minutes. From another study we know that the population standard deviation was 0.72 min. The 90% confidence interval for the true population mean time µ to polish a shoe is
A. (1.98, 2.42)
B. (1.95, 2.45)
C. (1.90, 2.50)
D. (1.99, 2.41)
E. none of the above
32. From a large approximately normal population 30 people are selected at random. If the sample mean age is 85.1 years and the sample standard deviation is 4.5 years, the 95% confidence interval for the true population mean is
A. (83.49, 86.71)
B. (83.46, 86.74)
C. (83.42, 86.78)
D. (83.39, 86.81)
E. none of the above
33. Find t*, the critical t value for a confidence level of 99% and a sample size of 17.
A. 2.898
B. 2.583
C. 2.921
D. 2.567
E. none of the above
34. The Labor Department wants to estimate the percentage of females in the U.S. labor force. They select a random sample of 525 employment records, and find that 210 of the people are females. The 90% confidence interval is
A. (0.3648, 0.4352)
B. (0.3581, 0.4419)
C. (0.3449, 0.4551)
D. (0.4235, 0.5679)
E. None of the above
35. If the Department of Labor wishes to tighten its interval, they should
A. Increase the confidence level.
B. Increase the sample size.
C. Decrease the sample size.
D. Both A and B
E. Both A and C
STAT 200 QUIZ 2
1. Which of the following are measures of central tendency? Select all that apply
a. mean
b. median
c. mode
d. Variance
e. standard deviation
f. linear transformation
2. Which of the following are measures of variability? Select all that apply
a. mean
b. median
c. mode
d. variance
e. standard deviation
f. linear transformation
3. Rachael and Peter are discussing how the mean value and variance affect the distribution graph. As the following graph shows, there are two distributions: A and B
Peter said that the distribution A has fatter tail because its has bigger standard deviation than the distribution B. Rachael's answer is different:she thinks the distribution A has a smaller standard deviation than the distribution B.
Who do you think is right?
a. Peter
b. Rachael
c. None of them are right. Distribution A and Distribution B have the same standard deviation
4. In a bell-shaped distribution, changing ___can make the distribution flatter (i.e. more compressed); and changing ___ can shift the distribution to the left or to the right.
a. Variance, mean
b. Mean, variance
c. Range, standard deviation
d. Mode, range
e. Mean, range
f. Mode, variance
5. A research study found that in one physics class, the correlation between pre-exam and post-exam grades was 0.90. The variance of pre-exam grades was 9 and the variance of post-exam grades was 16. Thus, the variance of the difference between pre-exam and post-exam grades is: (show work)
6. Image that a researcher examined how the people's heights are associated with their salaries. She found that the correlation between heights (measured in inches) and salary range (measured in dollars) was .36. Had she measured the height in cm instead, what would the correlation have been?
a. 0.36
b. 0.036
c. 3.6
d. 0.72&a