Exam 2 Review Guide
Math 150, FALL 2018
Directions: The exam will consist of 6 free response questions. All questions will be drawn from the questions written below. All answers on the exam must either have work shown or attached explanations. You are allowed one sheet of notes written on a 8.5x11 sheet of paper. You are also allowed the use of a calculator. It is HIGHLY recommended that you have a calculator for this exam.
The sections covered by the exam will be:
Correlation, Regression, Conditional Distributions, Probability, Counting Methods, Binomial Dis-tribution
Question 1) Suppose you do a sample in one city and collect the square footage of houses (in thousands of square feet) and the value of these homes (in thousands of dollars). You want to predict the value of a home y from its square footage x. Suppose there were 22 homes in your sample and all these homes were between 1200 and 2500 square feet.
You get the following regression line ˆy= 95x+ 115 and a correlation coefficient ofr =.82
(a) Give the value of the slope of this regression line. Interpret this value in the context of the situation. Does this make sense?
(b) Give the value of the intercept of this regression line. Interpret this value in the context of the situation. Does this make sense?
(c) Was there enough evidence in this case to claim a linear correlation? Why or why not?
(d) Give the value of the coefficient of determination. Interpret this value in the context of the problem.
(e) Suppose a house has 1500 square feet of space. What is its predicted value? Would you trust this prediction? Why or why not?
(f) Suppose you’re looking at a house that has 2000 square feet of space. It is currently on the market for 330,000 dollars. Is this a good deal according to your model?
Question 2) Suppose you are interested in studying the effectiveness of a typing class. To do this you randomly choose 8 people from the class and record how many weeks they’ve been taking the class and their current typing speed (measured in words per minute (wpm)).
Weeks in Class Typing Speed (wpm)
1 33
2 43
4 60
5 66
6 72
8 80
10 86
11 88
(a) Plot this data on a scatterplot.
(b) Calculate the correlation coefficient for this data. Is there enough evidence for a correlation? (c) Construct the least squares regression line for this data.
(d) Interpret the slope and intercept of this LSR. (e) Calcluate the residuals for this data.
(f) Plot the residuals.
(g) Is the LSR an appropriate model here? Why or why not?
Question 3) Suppose you run a small coffee shop and you want to study the relationship between daily temperature and number of cups of hot coffee sold at your shop. You record this information for a week and get the following data:
Day Temp ◦F Number of cups of Hot Coffee Sold
Monday 70 65
Tuesday 65 80
Wednesday 78 56
Thursday 76 58
Friday 80 64
Saturday 84 60
Sunday 72 76
(a) Calculate the correlation coefficient for this data. Is there enough evidence for a correlation? (b) Construct the LSR for this data.
(c) Interpet the slope and intercept for your LSR.
(d) Use the LSR to predict how many cups of hot coffee this shop would sell if the temperature was 75◦F. Would you trust this prediction?
(e) Use the LSR to predict how many cups of hot coffee this shop would sell if the temperature was 40◦F. Would you trust this prediction?
Question 4) Suppose you work for a company that wants to market a new pain reliever called Pain-B-Gone. To test the effectivness of this new drug you carry out an experiment where you have people who are experiencing headaches use either your new drug, a competing drug already on the market, or a placebo. Then you have them record if nothing happened, if their pain was slightly reduced or if their pain was entirely removed. You get the following data:
Removed Pain Entirely Reduced Pain Slightly Did Nothing
Pain-B-Gone 30 45 75
Competing Drugs 90 70 20
Placebo 10 25 65
(a) What was the size of your sample?
(b) Construct the conditional distribution based on the drug given.
(c) Which type of medicine resulted in the highest rate of the pain being removed entirely? What was that rate?
(d) Could you use this data to justify that Pain-B-Gone actually works? Why or why not?
(e) Could you use this data to justify that Pain-B-Gone is a better choice than other competing drugs? Why or why not?
Question 5) Consider the following probability experiment.
I get two fair six sided dice. I roll both of them and record the largest number I roll. However, if I roll the same number on both dice I record a 0. So if I roll both a 3 and a 4 I record a 4. If I roll a 5 and a 5 I record a 0.
(a) Write down a probability model for this experiment. That means write down all the possible outcomes and their associated probabilities. (Remember: outcomes here would be what I actually record)
(b) Calculate the probability you record an even number.
(c) Calculate the probability you record a number greater than 1.
(d) Calculate the probability you record an even number and a number greater than 1. (e) Calculate the probability you record an even number or a number greater than 1. (f) Calculate the probability you record a 5 given that you rolled a 2 on the first die.
Question 6) Consider the following data showing Ichiro Suzuki’s hits in the 2004 season of the MLB.
# of Hits Probability
0 .17
1 .33
2 .29
3 .15
4 .04
5 .02
(a) Verify this is a discrete probability distribution. (b) Draw a probability histogram for this variable. (c) What is the mean of this discrete random variable?
(d) What is the standard deviation of this discrete random variable?
(e) What is the probability that in a randomly chosen game in the 2004 season Ichiro had at least one hit?
(f) Assuming each baseball game is independent, what is the chance that Ichiro gets at least one hit in 5 games in a row?
Question 7) Suppose that a new president is being hired at El Camino. The final decision on the new hire is made by a randomly selected committee of 5 faculty from the Academic Senate. The Academic Senate consists of 11 STEM faculty, 10 Humanities faculty, 8 Social Sciences faculty, and 6 Health Science faculty.
(a) What is the probability that the committee consists entirely of STEM faculty?
(b) What is the probability that the committee consists of 2 STEM faculty, 2 Humanities faculty, and 1 Health Science faculty?
(c) What is the probability that the committee consists of entirely Humanities or Social Sciences faculty?
(d) What is the probability that the committee consists of exactly 2 STEM faculty? (e) What is the probability that the committee does not have any Social Science faculty?
(f) What is the probability that the committee consists of entirely STEM faculty given that the first two members selected are STEM faculty?
Question 8) Suppose you gather a random sample of 150 El Camino students and you ask them their favorite drink to have while studying and their favorite snack to have while studying. You get the following results:
Water Coffee Tea Soda/Energy Drinks
Chips 16 5 7 20
Candy 9 22 10 15
Fruit 17 9 16 4
(a) What is the probability that a randomly chosen student from this sample chose coffee as their favorite study drink?
(b) What is the probability that a randomly chosen student from this sample indicated that fruit was their favorite study snack?
(c) What is the probability that a randomly chosen student from this sample indiciated that water was their favorite study drink and that chips were their favorite study snack?
(d) What is the probability that a randomly chosen student from this sample indcated that tea was their favorite study drink or that candy was their favorite study snack?
(e) Are the events choosing water as your favorite study drink and fruit as your favorite study snack mutually exclusive? Why or why not?
(f) What is the probability that a randomly chosen student from this sample indicated that their favorite study snack was candy given that their favorite study drink was coffee?
(g) Are the events choosing soda/energy drinks as your favorite study drink and choosing chips as your favorite study snack indpendent or dependent? Why?
Question 9) Suppose that you have a box full of colored blocks of different shapes. Inside the box there are:
4 blue blocks which are a circle, a rectangle, a triangle, and a diamond. 6 green blocks which are three circles, two rectangles, and one triangles. 5 red blocks which are three rectangles, and two triangles.
Calculate each of the following probabilities assuiming there is no replacement when drawing ob-jects from the box
(a) P(1st draw is green) (b) P(1st draw is a circle)
(c) P(1st draw is red and a triangle) (d) P(1st draw is blue or a rectangle)
(e) P(1st draw is a rectangle given the 1st draw blue) (f) P(1st draw is green given the 1st draw is a diamond) (g) P(2nd draw is a circle given the 1st draw is a circle) (h) P(2nd draw is red given the 1st draw is green)
Question 10) Determine the number of different passwords for each of the following forms:
(a) Your password consists of 3 letters followed by 3 digits.
(b) Your password consists of 3 letters followed by 3 digits with no repeats allowed.
(c) Your password consists of 3 letters followed by 3 digits where you always start with aand end with the number 9 and you never repeat digits.
(d) Your password consists of 3 letters followed by 3 digits where all the letters are the same and all the digits are the same.
Question 11) Suppose that the chance that a randomly selected Math 150 student passes Math 150 is 63%. Further suppose each student is independent of one another. Suppose you are studying a randomly selected group of 12 Math 150 students.
(a) Find the probability that exactly 8 of these students pass Math 150. (b) Find the probability that at least 10 of these students pass Math 150. (c) Find the probability that less than 10 of these students pass Math 150. (d) Find the probability that exactly 5 of these students do not pass Math 150.
Question 12) Suppose that 54% of all internet searches are carried out by Google. Further suppose we examine a sample of 1000 internet searches. Treat this sample as a binomial random variable. (a) What is the mean of this random variable?
(b) What is the standard deviation of this random variable?
(c) Are the assumptions of estimating this random variable using the normal distribution met? Why or why not?
(d) What is the probability that in a sample of 1000 internet searches less than 500 of them were done by Google?
