STAT E100Exam 2 Review

Course Review

- Projects due at the final exam.- Exam 2 is Nov 26th, practice tests have already been posted.- Exams are cumulative, about 20% future exams will be old stuff.- The make-up exam will be held Thursday, Dec 5, 2013, 6-

7:30pm, in Science Center 309 for local test takers only. - There will not be a distance option for the make-up exam.- Don’t forget to review the homework solutions for the exam!- Email your TA to join the study group.

Key Concepts:

The binomial distribution is characterized by 4 properties:

1) Fixed number (n) of observations, or `trials’.

2) The n trial are all independent of each other

3) Each trial has two possible outcomes: `success’ or `failure’.

4) The probability (denoted by p) of success at each trial is constant.

Key Equations:

The main implications of the result

- The sampling distribution of is centered over the true population proportion, p.

- Note the formula of the standard deviation of .

- What happens as n increases?

- The standard deviation also depends on the unknown parameter p.

Key Inference Concepts and Equations:

Two main inferential techniques:

Confidence Intervals - for estimating values of population parameters

Hypothesis Testing- for deciding whether the population

supports a specific idea/model/hypothesis

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related. i) Which of the following will increase the width of a confidence interval for a population mean (assuming that everything else remains constant)? a) Decreasing the confidence levelb) Decreasing the sample sizec) Decreasing the standard deviationd) Decreasing the mean ii) The chamber of commerce in a beach resort town wants to estimate the proportion of visitors who are repeat visitors. From previous experience they believe the proportion is in the vicinity of 0.5 and they want to estimate the proportion to within ± 0.03 percentage points with 95 percent confidence. The sample size they should use is: a) n = 1068b) n = 545c) n = 33d) n = 95

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related. i) Which of the following will increase the width of a confidence interval for a population mean (assuming that everything else remains constant)? a) Decreasing the confidence levelb) Decreasing the sample sizec) Decreasing the standard deviationd) Decreasing the mean ii) The chamber of commerce in a beach resort town wants to estimate the proportion of visitors who are repeat visitors. From previous experience they believe the proportion is in the vicinity of 0.5 and they want to estimate the proportion to within ± 0.03 percentage points with 95 percent confidence. The sample size they should use is: a) n = 1068b) n = 545c) n = 33d) n = 95

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related. i) Which of the following will increase the width of a confidence interval for a population mean (assuming that everything else remains constant)? a) Decreasing the confidence levelb) Decreasing the sample sizec) Decreasing the standard deviationd) Decreasing the mean ii) The chamber of commerce in a beach resort town wants to estimate the proportion of visitors who are repeat visitors. From previous experience they believe the proportion is in the vicinity of 0.5 and they want to estimate the proportion to within ± 0.03 percentage points with 95 percent confidence. The sample size they should use is: a) n = 1068b) n = 545c) n = 33d) n = 95

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related. iii) In general, the smaller the p-value, the a) stronger the evidence against the alternative hypothesisb) stronger the evidence for the null hypothesisc) stronger the evidence against the null hypothesisd) none of the above  iv) A 95% confidence interval for the proportion of young adults who skip breakfast is 0.20 to 0.27. Which of the following is the correct interpretation of the 95% confidence interval? a) There is a 95% probability that the proportion of young adults who skip breakfast is between 0.20 and 0.27.b) If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.c) We can be 95% confident that the proportion of young adults in the sample who skip breakfast is between 0.20 and 0.27.d) We can be 95% confident that the proportion of young adults in the population who skip breakfast is between 0.20 and 0.27. 

i

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related. iii) In general, the smaller the p-value, the a) stronger the evidence against the alternative hypothesisb) stronger the evidence for the null hypothesisc) stronger the evidence against the null hypothesisd) none of the above  iv) A 95% confidence interval for the proportion of young adults who skip breakfast is 0.20 to 0.27. Which of the following is the correct interpretation of the 95% confidence interval? a) There is a 95% probability that the proportion of young adults who skip breakfast is between 0.20 and 0.27.b) If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.c) We can be 95% confident that the proportion of young adults in the sample who skip breakfast is between 0.20 and 0.27.d) We can be 95% confident that the proportion of young adults in the population who skip breakfast is between 0.20 and 0.27. 

i

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related. iii) In general, the smaller the p-value, the a) stronger the evidence against the alternative hypothesisb) stronger the evidence for the null hypothesisc) stronger the evidence against the null hypothesisd) none of the above  iv) A 95% confidence interval for the proportion of young adults who skip breakfast is 0.20 to 0.27. Which of the following is the correct interpretation of the 95% confidence interval? a) There is a 95% probability that the proportion of young adults who skip breakfast is between 0.20 and 0.27.b) If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.c) We can be 95% confident that the proportion of young adults in the sample who skip breakfast is between 0.20 and 0.27.d) We can be 95% confident that the proportion of young adults in the population who skip breakfast is between 0.20 and 0.27. 

i

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related.

v) Which one of these variables is a binomial random variable? a) time it takes a randomly selected student to complete a multiple choice examb) number of textbooks a randomly selected student bought this termc) number of women taller than 68 inches in a random sample of 5 womend) number of CDs a randomly selected person owns

Practice Exam A1. Multiple Choice: require no justification. Note: these parts are not related.

v) Which one of these variables is a binomial random variable? a) time it takes a randomly selected student to complete a multiple choice examb) number of textbooks a randomly selected student bought this termc) number of women taller than 68 inches in a random sample of 5 womend) number of CDs a randomly selected person owns

3. Three roommates live together. Let X be the random variable that measures the number of people that are home to eat dinner on any given night. Below is the probability distribution of X:

a. What is the mean of X?

b. What is the variance of X?

Practice Exam A

x 1 2 3

P(X = x) 0.2 0.3 0.5

3. Three roommates live together. Let X be the random variable that measures the number of people that are home to eat dinner on any given night. Below is the probability distribution of X:

a. What is the mean of X?

b. What is the variance of X?

Practice Exam A

x 1 2 3

P(X = x) 0.2 0.3 0.5

3.2)5.0(3)3.0(2)2.0(1)( xXxPX

3. Three roommates live together. Let X be the random variable that measures the number of people that are home to eat dinner on any given night. Below is the probability distribution of X:

a. What is the mean of X?

b. What is the variance of X?

Practice Exam A

x 1 2 3

P(X = x) 0.2 0.3 0.5

3.2)5.0(3)3.0(2)2.0(1)( xXxPX

61.0)5.0()3.23()3.0()3.22()2.0()3.21()()( 2222 xXPxX

3. Three roommates live together. Let X be the random variable that measures the number of people that are home to eat dinner on any given night. Below is the probability distribution of X:

c. Does X have a binomial distribution? How do you know?

d. Let be the random variable for the average number of roommates that are home to eat dinner over the course of n = 25 independent days. What is the approximate probability that is less than 2?

Practice Exam A

x 1 2 3

P(X = x) 0.2 0.3 0.5

3. Three roommates live together. Let X be the random variable that measures the number of people that are home to eat dinner on any given night. Below is the probability distribution of X:

c. Does X have a binomial distribution? How do you know?

No, X does not have a binomial distribution. In order for it to be binomial, there would have to be a chance of zero people eating at home, and this distribution does not have any probability that X = 0.

d. Let be the random variable for the average number of roommates that are home to eat dinner over the course of n = 25 independent days. What is the approximate probability that is less than 2?

Practice Exam A

x 1 2 3

P(X = x) 0.2 0.3 0.5

3. Three roommates live together. Let X be the random variable that measures the number of people that are home to eat dinner on any given night. Below is the probability distribution of X:

d. Let be the random variable for the average number of roommates that are home to eat dinner over the course of n = 25 independent days. What is the approximate probability that is less than 2?

Based on the CLT, we know that , so:

Practice Exam A

x 1 2 3

P(X = x) 0.2 0.3 0.5

)122.025/61.0/,(~ nNX XX

0069.0)46.2(122.0

3.22

/)2(

ZPZP

n

XZPXP

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

a) When 293 college students are randomly selected and surveyed, it is found that 114 own a car. Construct a 95% confidence interval for the percentage of all college students who own a car.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

a) When 293 college students are randomly selected and surveyed, it is found that 114 own a car. Construct a 95% confidence interval for the percentage of all college students who own a car.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

b) The table below shows the row and column marginal totals in a contingency table for two binary variables, A and B, each taking on values 0 or 1. Assume that the total sample size collected is fixed at 100. Fill in missing cell counts so that your values are consistent with the row and column totals and so that A and B are not independent (such that the χ2 test would definitely reject the null hypothesis). There is more than one correct answer. Briefly justify.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

b) The table below shows the row and column marginal totals in a contingency table for two binary variables, A and B, each taking on values 0 or 1. Assume that the total sample size collected is fixed at 100. Fill in missing cell counts so that your values are consistent with the row and column totals and so that A and B are not independent (such that the χ2 test would definitely reject the null hypothesis). There is more than one correct answer. Briefly justify.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

b) The table below shows the row and column marginal totals in a contingency table for two binary variables, A and B, each taking on values 0 or 1. Assume that the total sample size collected is fixed at 100. Fill in missing cell counts so that your values are consistent with the row and column totals and so that A and B are not independent (such that the χ2 test would definitely reject the null hypothesis). There is more than one correct answer. Briefly justify.

Under the null hypothesis, we would expect the values to all be 25 in the table. Any value further from this the better. A statistically significant example is shown above, but any values that replace the 40 in the table with a number between 30 and 50 in the table would be significant and receive full credit.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

c) In 2012 a research organization sent questionnaires to all the approximately 16,000 high schools in the United States. These questionnaires asked about iPad usage in the high school. About 4,000 high schools returned answers. Of these 4,000 high schools, 50% indicated that some of their students used iPads in class. In a recent speech, an authority on the use of iPads in high school education cited this study as evidence that "students in 50% of the high schools in the United States use iPads during their high school careers."

Do you regard 50% as a trustworthy estimate of the proportion of high schools providing iPad access in 2012? In two sentences or fewer, explain your answer.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

c)Do you regard 50% as a trustworthy estimate of the proportion of high schools providing iPad access in 2012? In two sentences or fewer, explain your answer.

No, 50% is not a trustworthy estimate of the true proportion of high schools that use iPads: only 1/4 of the schools responded leaving for a lot of potential for non-response bias. It is likely that schools that use iPads replied to this survey at a much higher rate.

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

Assume players on the PGA tour play only two different brands of golf balls: 70% play with Titleists, while 30% play with Nikes. What is the probability that 3 randomly selected players all play the same brand of ball?

Practice Exam A

Each part of this problem requires a short response with a brief explanation (simply yes or no will not suffice). Note: these parts are not related.

Assume players on the PGA tour play only two different brands of golf balls: 70% play with Titleists, while 30% play with Nikes. What is the probability that 3 randomly selected players all play the same brand of ball?

P(all 3 the same) = P(all 3 Titleist) + P(all 3 Nike) = 0.73 + 0.33 = 0.343 + 0.027 = 0.370

Practice Exam A