Descriptive
Stats
Stats
The heights of adult women are approximately Normally distributed about a mean of 65 inches, with a standard deviation of 2 inches. If Rachel is at the 99th percentile in height for adult women, then her height, in inches, is closest to?
A) 60 B) 62 C) 68 D) 70 E) 74
D) 70
When testing water for chemical impurities, results are often reported as bdl, that is, below detection limit. The following are the measurements of the amount of lead in a series of water samples taken from inner-city households (in parts per million):
5, 7, 12, bdl, 10, 8, bdl, 20, 6
Which of the following statements can we be sure is true?
(a) The mean lead level in the water is about 10 ppm.
(b) The mean lead level in the water is about 9 ppm.
(c) The median lead level in the water is 7 ppm.
(d) The median lead level in the water is 8 ppm.
(e) Neither the mean nor the median can be computed because some values are unknown.
C) The median lead level in the water is 7 ppm.
A dietary supplement manufacturer wants to test consistency of the krill oil content of omega-3 supplements produced in one factory. The company decides to select one bottle of capsules randomly from each day’s production run at the factory and analyze every capsule in this bottle as to krill oil content. What type of sampling is this?
A) Cluster Sampling
B) Convenience Sampling
C) Simple Random Sampling
D) Stratified Sampling
E) Systematic Sampling
A) Cluster sampling
The two-way table below gives information on seniors and juniors at a high school and by which means they typically get to school.
Car Bus Walk Totals
Juniors 146 106 48 300
Seniors 146 64 40 250
Totals 292 170 88 550
You select one student from this group at random. If the student says he is a junior, what is the probability that he walks to school?
(a) 0.073 (b) 0.160 (c) 0.455 (d) 0.600 (e) 0.833
B) 0.16
The report of a sample survey of 1,014 adults says, "With 95% confidence, between 9% and 15% of all Americans expect to spend more money on gifts this year than last year." What does the phrase "95% confidence" mean?
(a) 95% of all Americans will spend between 9% and 15% more than what they spent last year.
(b) 9% to 15% of all Americans will spend 95% of what they spent last year.
(c) there is a 95% chance that the percent who expect to spend more is between 9% and 15%.
(d) the method used to get the interval from 9% to 15%, when used over and over, produces intervals which include the true population percentage about 95% of the time.
(e) we can be 95% confident that the method used to get the interval always gives the right answer.
D
Students at a local high school have GPAs that are Normally distributed. It was thought that girls had higher GPAs than the boys. You take a random sample of 39 boys and find that they have an average GPA of 3.2 with a standard deviation of 0.8. You also sample 41 girls and find that their mean GPA is 3.6 with a standard deviation of 1.6. NAME THE INFERENCE PROCEDURE we could use to determine if girls had higher GPAs than boys!
2 sample t-test
The five-number summary for scores on a statistics exam is 11, 35, 61, 70, 79. In all, 380 students took the test. About how many had scores greater than 70?
(a) 26 (b) 76 (c) 95 (d) 190 (e) None
(C) 95
In a statistics course, a linear regression equation was computed to predict the final-exam score from the score on the first test. The equation was
predicted y = 10 + 0.9x
where y is the final exam score and x is the score on the first test. Bill scored a 90 on the first test and a 93 on the final exam. What is the value of his residual?
(a) –2.0 (b) 2.0 (c) 3.0 (d) 93 (e) none of these
(B) 2.0
A study was conducted to evaluate the impact of taking a nutritional supplement on a person’s reaction time. One hundred volunteers were placed into one of three groups according to their athletic ability: low, moderate, or high. Participants in each group were randomly assigned to take either the nutritional supplement or a placebo for six weeks. At the end of the six weeks, participants were given a coordination task. The reaction time in completing the task was recorded for each participant. The study compared the reaction times between those taking the supplement and those taking the placebo within each athletic ability level. Which of the following is the best description of the study?
(A) A completely randomized design
(B) A matched-pairs design
(C) A randomized block design
(D) A randomized observational study
(E) A stratified observational study
C) A randomized block design
A die is loaded so that the number 6 comes up three times as often as any other number. What is the probability of rolling a 1 or a 6?
(a) 2/3 (b) 1/2 (c) 3/8 (d) 1/3 (e) 1
(B) 1/2
You are going to create a 95% confidence interval for a population proportion and want the margin of error to be no more than 0.05. Historically, data indicated that the population proportion has remained constant at about 0.7. What is the minimum sample size needed to create this confidence interval?
324
What do we call the value that is the square of the standard deviation?
Variance
Suppose that the distribution for a set of scores has a mean of 47 and a standard deviation of 14. If 4 is added to each score, what will be the mean and the standard deviation of new scores?
A) Mean = 51 SD = 14
B) Mean = 51 SD = 18
C) Mean = 47 SD = 14
D) Mean = 47 SD = 16
E) Mean = 47 SD = 18
A) Mean = 51 SD = 14
Find & interpret (a)correlation coefficient and (b) slope for predicting height from shoe size.
Predictor Coef SE Coef T P
constant -4.792 8.521 -0.56 0.594
height 0.6077 0.1236 4.92 0.003
S = 0.9323 R-Sq = 80.1% R-Sq(adj) = 76.8%
(a) There is a strong positive association between shoe size and height.
(b) For every additional unit of shoe size, the predicted height should increase by 0.60 in.
We wish to draw a sample of 5 without replacement from a population of 50 households. Suppose the households are numbered 01, 02, . . . , 50, and suppose that the relevant line of the random number table is 11362 35692 96237 90842 46843 62719 Then the households selected are
(a) households 11 13 36 62 73
(b) households 11 36 23 08 42
(c) households 11 36 23 23 08
(d) households 11 36 23 56 92
(e) households 11 35 96 90 46
(B) households 11 36 23 08 42
Suppose a computer chip manufacturer rejects 2% of chips produced because they fail pre-sale testing.
What is the probability that the fifth chip you test is the first bad one you find?
0.0184
Two Polish math professors and their students spun a Belgian Euro coin 250 times. It landed on heads 140 times. One of the professors concluded that the coin was minted asymmetrically. A representative from the Belgian mint said that the result was just by chance. Is the math professor or the rep from the Belgian mint correct? Calculate the appropriate p-value for this scenario.
0.0287, which means the coin landing on heads 140 out of 250 times is extremely rare if this is a fair coin. This leads to a conclusion that the professor was correct in his thinking!
A survey of 430 randomly selected adults found that 21% of 222 men and 18% of 208 women had purchased books online. Is there evidence that men are more likely than women to make online purchases of books? Write the hypotheses, defining any parameters used and state the name of the test you would use to answer the question.
Pm = the true proportion of men who purchase books online
Pw = the true proportion of women who purchase books online
Ho: Pm = Pw Ha: Pm > Pw
Two proportion z-test
Vanessa is enrolled in a very large college Calculus class. On the first exam, the class mean was a 75 and the standard deviation was 10. On the second exam, the class mean was 70 and the standard deviation was 15. Vanessa scored 85 on both exams. Assuming the scores on both exams were approximately Normally distributed, on which exam did Vanessa scored better, relative to the rest of the class?
A) It is impossible to tell because the class size is not given.
B) It is impossible to tell because the correlation between the two sets of exams is not given.
C) She scored much better on the first exam.
D) She scored much better on the second exam.
E) She scored about equally well on both exams.
E) She scored about equally well on both exams.
The table below shows the relationship between student enrollment (in thousands) and total number of property crimes (burglary and theft) in 2006 for eight colleges and universities in a certain U.S. state.
Enrollment (in 1000s) # of Property Crimes
16 201
2 6
9 42
10 141
14 138
26 601
21 230
19 294
Draw the scatterplot for this data on your calculator and describe the association between enrollment and number of property crimes.
There is a positive, roughly linear, moderately strong association between the number of students enrolled and the total number of property crimes. There is a possible outlier at 26,000 students and 601 crimes.
As a researcher for a pharmaceutical company, you are designing a study to test the effectiveness of a new treatment for migraine headaches. You have been given a list of 126 people willing to participate in the trial. The first 70 people are female; the remaining 56 are male. Preliminary research suggests that men and women respond differently to this new treatment. What sort of experimental design would you choose for this study, and why?
Randomized block design by first dividing the subjects into blocks of males and females. This allows for comparison between the effectiveness of the medication without the confounding variable of gender.
Wile E. Coyote is pursuing the Road Runner across Great Britain toward Scotland. The Road Runner chooses his route randomly, such that there is a probability of 0.8 that he’ll take the high road and 0.2 that he’ll take the low road. If he takes the high road, the probability that Wile E. catches him is 0.01. If he takes the low road, the probability he gets caught is 0.05. Find the probability that he took the high road, given that he was caught.
0.44
When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution mu = with 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded.
Describe a Type I and a Type II error in this situation and the consequences of each.
Type I We are convinced that the true mean lifetime of the batteries is less than 7 hours, when actually the lifetime is greater than that. Perfectly good batteries will be discarded!
Type II We are not convinced that the true mean lifetime of the batteries is less than 7 hours, when actually it is. Batteries will be sold that will last much shorter time than customers expect.
Are all employees equally prone to having accidents? To investigate this hypothesis, a researcher looked at a light manufacturing plant and classified a random sample of accidents by type and by age of the employee. Age Sprain Burn Cut
Under 25 9 17 5
25 or over 61 13 12
Here are the expected counts for this table under the null hypothesis that tests the question stated above.
18.5 7.9 4.5
51.5 22.1 12.5
What is the individual component of the chi-square statistic for the cell “Under 25/Burn?”
(a) 1.15 (b) 4.87 (c) 5.64 (d) 9.10 (e) 10.48
E
In some courses (but certainly not in an intro stats course!), students are graded on a “Normal curve.” For example, students within ± 0.5 standard deviations of the mean receive a C; between 0.5 and 1.0 standard deviations above the mean receive a C+; between 1.0 and 1.5 standard deviations above the mean receive a B–; between 1.5 and 2.0 standard deviations above the mean receive a B, etc. The class average on an exam was 60 with a standard deviation of 10. The bounds for a B– grade and the percent of students who will receive a B–grade if the marks are actually Normally distributed are
(a) (65, 75), 24.17%
(b) (65, 75), 12.08%
(c) (70, 75), 18.38%
(d) (70, 75), 9.19%
(e) (70, 75), 6.68%
D) (70, 75), 9.19%
Data is available that explores the relationship between latitude and average July temperature in the twelve largest U.S. cities. The value of r-squared for these data is 0.277. Interpret this value in the context of the problem.
27.7% of the variability in predicted July temperatures is accounted for by the LSRL between latitude and July temps.
A church group interested in promoting volunteerism in a community chooses an SRS of 200 community addresses and sends members to visit these addresses during weekday working hours to inquire about the residents’ attitudes toward volunteer work. Sixty percent of all respondents say that they would be willing to donate at least an hour a week to some volunteer organization. Bias is present in this sample design. Explain the bias involved in this problem.
Sampling only during workday hours meant that only people without regular daytime jobs were available to answer the door— the poll suffered from undercoverage of people who were employed. Since those who are not employed may be more likely to have time to volunteer, the poll probably overestimated the actual proportion of potential volunteers.
There is also potential response bias: a people is likely to say he or she will volunteer to look like a good person. This would also result in an overestimation of the actual proportion of potential volunteers.
A pharmaceutical lab claims that a drug it produces causes serious side effects in 20 out of every 100 people. To check this claim, a hospital administers the drug to 15 randomly selected patients and finds that at least 5 suffer serious side effects. If the lab’s claims are correct, what is the probability of the hospital obtaining the results it did?
0.164
Nicotine patches are often used to help smokers quit. Does giving medicine to fight depression also help? A randomized double-blind experiment assigned 244 smokers to receive nicotine patches and another 245 to receive both a patch and the antidepressant drug bupropion. A 99% confidence interval for the difference in the proportion of smokers who abstain when using buproprion and a nicotine patch and the proportion who abstain when using only a patch is
(0.091, 0.291)
A hypothesis test at the alpha = 0.01 significance level would have what conclusion?
We reject Ho. There is convincing evidence that the proportion of smokers who abstain when using buproprion and the patch is greater than the proportion who abstain when using the patch only.
What conditions must be met for a random variable to be considered binomial?
2 outcomes, success and failure
fixed # of trials
independent trials
probability of success is the same for all trials