Regression, Combinatorics, Probability,Name:

and Inference Review

1.Association of measurement variables

  • Describe the relationship between two variables (measurement v measurement)
  • Create and interpret scatter plots
  • identify outliers

1.Open the dataset “SAT by State 06.xls.” Construct a scatterplot of “math” versus “percent taking”

(a.)positive, strong, linear

(b.)negative, moderate, linear

(c.)negative, strong, non linear

(d.) no relationship

2.Open the dataset “Governors05.xls.” Construct a scatterplot of “Governor Salary” versus “Median house value.” Which best describes the relationship?

(a.)positive, strong, linear, with no outliers

(b.)negative, moderate, linear, with an outlier

(c.)positive, moderate, linear with an outlier

(d.) no relationship

3.In the scatterplot of “Governor Salary” versus “Median house value,” which state is an outlier?

(a.)Pennsylvania

(b.)Hawaii

(c.)California

(d.) Alaska

2. Correlation Coefficient

Calculate and interpret the correlation coefficient

4.Calculate the correlation between “Median Housing Value” and “Governor Salary.”

(a) -.02

(b) .55

(c) .33

(d) -.55

5.Which of the following is true about correlation?

(a.)it is only appropriate to use correlation when the relationship is linear

(b.)if the correlation coefficient is close to 1 or -1, there is a cause-and-effect relationship

(c.)correlation is resistant to outliers

(d.) all of the above are true

6. If a correlation coefficient between Age and Coolness Rating is -0.70, then:

(a) A person’s Age is usually less than the person’s Coolness Rating.

(b) A person’s Age is usually more than the person’s Coolness Rating.

(c) Below average Ages tend to be associated with below average values of Coolness.

(d) Below average Ages tend to go with above average values of Coolness.

3. Calculate and Apply Regression Equation

  • Calculate the regression equation
  • Graph and make predictions based on the LSR line
  • Understand the difference between an outlier and an influential observation in the context of regression – level 1 only

7.“SAT by State 06.xls.” Use StatCrunch to calculate the linear regression equation of “Math” versus “Percent Taking”

Equation:______

8. Use the regression equation to predict the Math SAT score for a state with 50% of students taking the SAT

Answer:______

9.What is the y-intercept of the equation?

Answer: ______

4. Interpret Regression Equation

  • Interpret the slope and intercept of the regression equation
  • Understand the Least Squares criterion for line of best fit

10.Explain what the slope of the regression equation (in 8.) means in context.

11.Below is the Statcrunch output for a regression run on “Airfare” (in dollars) versus “Distance” (in miles) for a set of data for flights between 12 cities.

Simple linear regression results:
Dependent Variable: Airfare
Independent Variable: Distance
Airfare = 83.26736 + 0.11737509 Distance
Sample size: 12
R (correlation coefficient) = 0.795
R-sq = 0.63200194
Estimate of error standard deviation: 37.827023

The slope of the regression equation indicates that…

(a)There is a strong negative linear relationship

(b)An additional mile tends to cost about 11.7 cents

(c)A trip of 0 miles costs .117 dollars

(d)None of the above

12.The “line of best fit” under the least squares criterion is the one which…

(a)minimizes the sum of all the perpendicular distances from the points to the line.

(b)has the greatest sum of squared errors in predictions.

(c)has the smallest sum of squared residuals

(d)has the smallest average residual

5. Transforming Data

  • Apply a transformation of the data to create a more appropriate model
  • use a residuals plot to evaluate the appropriateness of the model

13.It may be necessary to transform the data such as by computing logs or square roots on before performing a linear regression if…

(a)there are many outliers

(b)the scatterplot shows a nonlinear relationship

(c)there is no apparent relationship between the variables

(d)all of the above

14.Use the data set “SAT by State 06.xls.” Construct a scatterplot of “Writing” versus “Percent Taking.” Verify that the scatterplot shows a curved pattern. Transform the data by computing Sqrt(Percent Taking). Compute a linear regression of Distance versus Sqrt(Percent Taking). Write the regression equation below.

Answer: ______

15.Construct a residuals plot for the regression computed in 14. What does it show? What does it mean?

6. Basic counting rules

  • Use the multiplication and addition rules of counting
  • Compute and apply factorials
  • Construct and apply tree diagrams and tables

16.If Mr. Mellor has 4 shirts and 3 pairs of pants, how many different outfits can he put together? Make a tree diagram to show all the possible outfits.

17.How many 3-digit numbers contain no 4’s? How many 3-digit numbers contain at least one 4?

18.How many ways can 5 T/F questions be answered?

19.How many ways can the letters in the word “COMPUTER” be arranged?

7. Advanced counting rules

  • Use combinations and permutations to solve counting problems
  • Distinguish between situations where combinations, permutations, or other counting rules apply

20.Three door prizes are to be given to 3 lucky people in a crowd of 100.

a.If the three prizes are identical, in how many ways can this be done?

b.If the three prizes are different (1st , 2nd, and 3rd), in how many ways can this be done?

21.Baskin Robbins has 31 flavors. How many three scoop cones can be made if order is important and flavors cannot be repeated?

22.How many three scoop cones are possible if flavors can be repeated?

8. Law of Large Numbers

  • Understand the relative frequency interpretation of probability and the Law of Large Numbers
  • Conduct simulations with physical models and random number generators to make empirical estimates of probability

23.Which is a correct interpretation of the Law of Large Numbers?

(a.)If you flip a coin 10 times and get 9 heads, we expect to get tails on the next toss.

(b.)If you flip a coin a small number of times you are more likely to get close to 50% “heads” than if you flip a coin a large number of times.

(c.)If you flip a coin twice and get “heads” both times, the next flip will most likely come up “tails”

(d.)If you flip a coin 50 times, you probably won’t get exactly 25 “heads” but you are likely to get somewhat close to 50% “heads”

Standard 9. ComputingProbabilities

  • Compute probabilities using an appropriate sample space
  • Compute probabilities using counting rules including permutations and combinations

24.Which of the following is NOT a possible probability?

(a.) 25/100

(b.) 1.25

(c.) 1

(d.) 0

25.When rolling a pair of 6-sided dice, what is the probability of rolling a sum of 6 or 7?

(a.) 1/6

(b.) 3/14

(c.) 5/12

(d.) 11/36

26.Tina has 5 red, 6 blue, 3 white, and 4 orange marbles. All marbles are put in a sack and one marble is selected at random. Compute the probability of drawing a red marble and the probability of drawing a blue or white marble.

(a.) 4/19; 9/20

(b.) 3/10; 11/20

(c.) 5/18; 1/2

(d.) 9/21; 7/18

27.A statistics project was assigned in a class of 20 students. Each student will present his or her work to the class. Three of the 20 students will be selected at random each day to present to the class, and presentations will continue over several days. What is the probability that you and your two best friends in the class will be the 3 students selected to present on the very first day of project presentations?

(a.) 1/1140

(b.) 3/20

(c.) 1/2130

(d.) 2/101

Standard 10. Compound Events

  • Solve compound probability problems using tree diagrams and area diagrams
  • Use conditional probabilities – level 1

28.In a basketball one-and-one situation. If a player makes the first foul shot, she gets another foul shot. If she misses it, she does not get to shoot again. (So a player can score 0,1, or 2 points in a one-and-one situation.) Suppose Liz has a 70% chance of making each foul shot. What is the probability that she scores exactly one point in a one-and-one situation?

(a.) 70%

(b.) 9%

(c.) 49%

(d.) 21%

29.A drawer contains 5 black socks and 3 blue socks. If you reach into the drawer without looking and pull out two socks, what is the probability that you get a matching pair?

(a.) 21/56

(b.) 13/28

(c.) 4/56

(d.) 1/2

30.What is the probability that 3 students selected at random were all born on different days of the week? What is the probability that at least two of them were born on the same day of the week?

(a.) 30/49; 19/49

(b.) 36/49; 2/49

(c.) 30/56; 25/36

(d.) 2/3; 1/3

Standard 11. Expected Value

  • Compute expected value
  • Make and defend decisions based on expected value calculations

31.Use the table to compute the expected value for X.

X12-3

P(X).3.2.5

(a.)1.2

(b.) 2.5

(c.)-.8

(d.) -2

32.A die is rolled. If the number rolled is odd, Player A wins $2 from Player B. If it is a 6, A wins $4 from B. Otherwise B wins $5 from A. Is this a fair game?

(a.)Yes, because the expected value is 0.

(b.) Yes, because the expected value is positive for player A

(c.)No, because the expected value is negative for player A

(d.) No, because the expected value is 0

33.Refer to Liz’s one-and-one situation in number 5. What is Liz’s expected value for the average number of points she scores in a one-and-one situation?

(a.)1.2

(b.) 2.5

(c.)-.8

(d.) -2

Standard 12. Using the binomial distribution

  • Graph a binomial distribution
  • Compute probabilities based on a binomial distribution

34.Which of the following sets of values when substituted into the formula above gives the probability of getting 6 sixes in 10 rolls of a die?

(a) n = 4, p = 10, k = 1/6

(b) n = 10, p = 1/6, k = 4

(c) n = 10, p = 6, k = 1/6

(d) n = 4, p = 1/6, k = 10

35.Which of the following is not a situation where the binomial probability formula should be applied?

(a.)A coin is flipped 20 times. What is the probability of getting 5 or more heads?

(b.) What is the probability that the first 6 occurs on the 5th roll of a six-sided die?

(c.)What is the probability that Liz will make exactly 3 out of 5 shots if she has a 70% chance of making each shot and we assume independence?

(d.) A bag contains 6 marbles of which two are green. A marble is selected at random from the bag, the color is noted, and the marble is returned to the bag. Repeating this action 15 times, what is the probability of getting 5 or fewer green marbles?

36.The Good Guys have won 60% of their games over a long season. With seven games to go, what is the probability that the Good Guys will win at least five of them? Assume that the Good Guys have a .6 probability of winning each game.

(a.).6

(b.) .36

(c.).42

(d.) .48

Standard 13. Using Normal Distributions

  • Sketch the graph of a normal distribution
  • calculate z-scores
  • Calculate probabilities/proportions pertaining to a normal distribution
  • Calculate values of a variable corresponding to given probabilities or proportions
  • Apply the normal approximation to the binomial distribution - level 1 only

37.The distribution of lifetimes for a certain type of light bulb is normally distributed with a mean of 1000 hours and a standard deviation of 100 hours. Find the 33rd percentile of the distribution of lifetimes.

(a.) 560

(b.) 330

(c.) 1044

(d.) 1440

(e.) none of these

38.Compute the z-score for a score of 70 on a test with the following summary statistics:

min = 20, Q1 =70, med = 75, Q3 = 76, max = 100, s = 4, = 70

(a) z = 2

(b) z = 2.5

(c) z = 0

(d) z = 7

39.A negative z-score indicates…

(a) that someone made a computational error

(b) that the mean is a negative number

(c) that the distribution is skewed toward lower values

(d) that the item in question is below the mean

40.A person with a Z-Score of -2.00 has performed below approximately what percent of the students taking the test?

(a) 2 percent

(b) 15 percent

(c) 50 percent

(d) 84 percent

41.In a population there are 60 scores, the mean is 45 and a standard deviation is 25. A score of 85 from this population:

(a.) has a Z-score of 1.60.

(b.) has a Z-score of 1.00.

(c.) has a Z-score of -1.00.

(d.) it is impossible to compute Z without additional information

42.One hundred students took a test on which the mean score was 73 with a standard deviation of 8. A grade of A was given to all who scored 85 or better. Approximately how many A's were there, assuming scores were normally distributed? (Choose the closest.)

(a.) 42

(b.)7

(c.)58

(d.)5

(e.)22

Standard 16. Carrying out statistical tests

  • Carry out a test of significance (hypotheses, collect and organize data, compute test statistic, compute p-value, make decision, interpret in context)

43.Suppose a medical experiment randomly assigns patients with a certain condition to either receive an “old” treatment or a “new” treatment. The experimenters are interested in whether the new treatment works better than the old treatment. Which of the following would be an appropriate null hypothesis?

(a.)The new treatment works better than the old treatment

(b.)The new treatment works the same as the old treatment

(c.)The old treatment works better than the new treatment

(d.)Neither the old or new treatments work.

44.Medical experimenters randomly assigned 164 pregnant HIV-positive women to receive the drug AZT during during pregnancy while another 160 such women were randomly assigned to a control group which received a placebo. Of those in the AZT group, 13 had babies who tested HIV-positive, compared with 40 HIV-positive babies in the placebo group. Contruct two-way tables of observed and expected counts. If there is no difference between AZT and the placebo, what number of babies of mothers in the control group would be expected to be HIV positive?

(a.) 26.2 (b.)19.3

(c.)40(d.)13

45.In an experiment to study attempting to identify factors which can influence people’s responses to survey questions, subjects were randomly given one of the two following statements and asked whether they agree or disagree:

A: “Individuals are more to blame than social conditions for crime and lawlessness in this country.”

B: “Social conditions are more responsible than individuals for crime and lawlessness in this country.”

Blame Individuals / Blame Social Conditions / Total
A / 282 / 191 / 473
B / 204 / 268 / 472
Total / 486 / 459 / 945
Statistic / DF / Value / P-value
Chi-square / 1 / 25.4347 / <0.0001

The responses are summarized below:

Which of the following is an appropriate conclusion for a chi-square test of the null hypothesis that the wording of the question did not influence the responses:

(a.) Accept the null hypothesis. There isn’t sufficient evidence to demonstrate that the wording of the question had an effect.

(b.)Reject the null hypothesis. There is strong evidence that people were influenced by a tendency to agree with the researcher

(c.)Fail to reject the null hypothesis. People were more likely to disagree than to agree with whichever statement was given.

(d.)Reject the alternative hypothesis. There no evidence suggesting that the wording of the question influenced responses.

Standard 17. Understanding Statistical Tests

  • Explain p-value as a probability
  • Understand the reasoning for a test of significance

46.Which of the following is the best explanation of a null hypothesis, Ho, in a statistical test.

(a) a statement of what we are trying to prove in an experiment

(b) it is usually a statement that “there really is a difference”

(c) a neutral assumption made for the sake of argument that the researcher tries to disprove based on conflict with the observed data

(d) it is the hypothesis that the data cannot be explained by random variability

47.A p-value of 0.11 would indicate…

(a) that someone made a computational error

(b) strong evidence against the null hypothesis

(c) that the alternative hypothesis must be false

(d) that we should fail to reject the alternative hypothesis at a .05 significance level

48.Roberto suspects that his brother’s coin is biased towards heads. He flips the coin a large number of times and records the results. Suppose that the number of “heads” he got differs from the expected number of “heads” from a fair coin so much that the p-value in a Chi-square test is 0.04. Which of the following is not a correct interpretation of this p-value?

(a) In only 4 random samples in one hundred would we see such a large difference between the observed and expected numbers of heads when flipping a fair coin

(b) The null hypothesis of the test is rejected at the .05 significance level

(c) The coin is biased such that it comes up “heads” less than 4% of the time.

(d) If Roberto’s brother’s coin is fair, the probability of observing such a large difference between the observed and expected numbers of “heads” is fairly small.

49.It is appropriate to reject the null hypothesis in a test of significance if…

(a) it would be very unlikely to get the sort of data that was observed if the null hypothesis were true.

(b) the data are consistent with what would be expected if the null hypothesis were true

(c) the significance level if .05 and the p-value is greater than .05

(d) the chi-square value is close to 0

Standard 20. Experimental Design

  • Understand principles of control including comparison, replication, randomization, blindness, and blocking
  • Carry out a well-controlled experiment

50.What is the role of randomization in an experiment?

(a.) A random sample of subjects is selected from a population to avoid bias

(b.) Subjects are randomly assigned to treatments so create experimental groups that are similar

(c.) Treatments are applied to every subject at random intervals

(d.) Experimenters must randomly decide what the purpose of the study is

51.The control group in an experiment should be designed to receive:

(a.) the opposite of the experiences afforded the experimental group.

(b.) the same experiences afforded the experimental group except for the

treatment under examination.

(c.) the experiences afforded the experimental group except for

receiving the treatment at random intervals.

(d.) the experiences which constitute an absence of the experiences

received by the experimental group.

52.In an experiment to see if Smartfood Popcorn makes people smarter, 50 students are randomly assigned to two groups. The first group receives Smartfood popcorn and the second gets regular popcorn. Then each student writes an essay on an assigned topic. The essays are scored by a teacher and the results are compared for the two groups to see if Smartfood had a positive effect on essay writing. Which of the following would ensure blindness and double-blindness in this study?

(a.) the students and reseachers should both wear blindfolds

(b.) no one should know whether Smartfood was used for either group.

(c.) the students should not know whether they got Smartfood or a different brand of popcorn, and the teacher should not know which group the the student was in while grading the essay.

(d.) Smartfood should be given to both groups but in random intervals so that the students do not know when they are getting it

Standard 21.Studies

  • distinguish between explanatory and response variables
  • distinguish between observational studies andexperiments
  • understand when a "cause and effect" conclusion is justified
  • understand and identify lurking (confounding) variables

53..An experiment is conducted to determine if the use of certain specified amounts of a drug will increase IQ scores. In this study, IQ serves as:

(a) an explanatory variable