# I am interested in seeing if income - Expert Answers

Question 11) I am interested in seeing if income influences opinions about the legalization of marijuana. I take a random sample of 655 people who make above 100 thousand a year and ask if they support the legalization of marijuana. 352 say that they do. I take another random sample of 600 people who make below 100 thousand a year and ask if they support the legalization of marijuana. 350 say that they do.

Carry out a hypothesis test at α = .05 to see if income causes a difference in opinions about the legalization of marijuana. If there is significant evidence construct a 95% confidence interval for the difference.

Question 12) Suppose I want to study the amount of time people spend watching TV on a weekday. I find that the average time is 2.35 hours with a standard deviation of 1.93 hours.

(a) What shape do you expect this distribution to have? Why?

(b) Suppose now I take a sample of size 100. Describe the distribution of the sample mean.

(c) What is the probability that in my sample the average time spent watching TV is between 2 and 2.5 hours?

(d) What is the probability that in my sample the average time spent watching TV is below 1.75 hours?

(e) If you observed the results mentioned in (d) what might you conclude?

Question 13) Suppose that a car company claims their new compact car gets 30 miles per gallon. A consumer advocacy group worries that this new compact car gets less than this. They take a random sample of 15 cars and test their fuel efficiency. They get the following data, in miles per gallon.

(a) Carry out a hypothesis test at α = .05 to see if there is evidence for the advocacy group’s claim. Make sure to confirm all necessary conditions.

(b) Regardless of your answer to (a) build a 95% confidence interval for the average miles per gallon that this type of car gets.

(c) If the advocacy group is worried of falsely accusing the car company of producing cars that get less than 30 miles per gallon should they retest at α = .01 or α = .1?

(a) Build a 90% confidence interval for the true proportion of how often this slot machine pays out. Make sure to use an appropriate method and confirm all necessary conditions.

(b) Using your interval would you consider it be likely, unlikely, or impossible that the true proportion is 2%?

(c) Using your study as a pilot study, how many times would you need to play the machine to build a 90% confidence interval with a margin of error of less than 1%?

Question 15) Suppose you are interested in studying if caffeine decreases reaction time. To do this you take a random sample of 10 people. You bring them to your lab one day and give them a reflex test. The reflex test is done by putting two buttons in front of them, one that says red and one that says blue. A color then flashes on the screen and the subject has to press the correct button. You then bring all the participants back the next day and give them caffeine pills. You wait twenty minutes for the pills to take effect and then give them the test again. You get the following results (note all values are in milliseconds):

(a) Carry out a hypothesis test at α = .01 to test the researcher’s claim. Make sure to confirm all necessary conditions.

(b) Regardless of your answer to (a), build a 99% confidence interval for the difference in reaction time. Make sure to write a sentence interpreting your interval.

(c) Are individual differences in reaction time a flaw in this experiment? Why or why

(d) Give one major flaw in this experiment. How could this be corrected?

Question 16) Suppose you work for a fast food company. You are interested in studying if people spend different amounts when going through the drive-thru versus actually ordering inside the restaurant. You take a random sample of 81 bills from the drive thru and find that the average amount spent was 9.38 with a standard deviation of 3.74. You also take a random sample of 61 bills from dine-in customers and find that the average amount spent was 10.57 with a standard deviation of 4.31.

(a) Carry out a hypothesis test at α = .05 to test this claim. Make sure to confirm all necessary conditions.

(b) Regardless of your answer to (a), construct a 95% confidence interval for the average

(c) Describe what it would mean to make a Type II error in this situation.

Question 17) Suppose you are interested in studying the relationship between college major and whether students enjoy the math classes they take. You take a sample of 150 students from a local university and ask them to identify their major and say if they have enjoyed, felt neutral about, or not enjoyed the math classes they have taken so far. You get the following data:

(a) If you randomly chose someone from your study what is the chance their major was in the social sciences?

(b) If you randomly chose someone from your study what is the chance that their major

(c) If you randomly chose someone from your study what is the chance that their major

(d) If you randomly choose someone from your study what is the chance they enjoyed their math courses given they were in the Arts and Humanities?

(e) Which major had the highest rate of students that felt neutral about the math courses they had taken?

(a) Carry out a hypothesis test at α = .05 to test if there is a relationship between major and enjoyment of math classes. Make sure to confirm all necessary conditions.

Question 18) For this question use the data from Question 18

(b) Describe what it would mean to make a Type I error here.

(c) Describe what it would mean to make a Type II error here.

Question 19) Suppose you work for a large town government. You are studying the ethnic diversity of your area. From previous research in 2012 you had the following ethnic breakdown:

(a) Test at α = .05 to see if the ethnic diversity in your area has changed since 2012. Make sure to check all necessary conditions and state your conclusion in the context of this situation.

(b) If you were worried about missing a change in the ethnic diversity of your area would you retest at α=.01or α=.10? Why?

Question 20) Suppose you are interested in studying the average number of hours students study the night before an exam. You take a random sample of 100 students from a local university all of whom are majoring in one of the STEM fields. After asking them to estimate how many hours they usually study the night before an exam you get the following 95% confidence interval in hours, (.25, 4.25).

Decide if the following interpretations are true or false.

(a) 95% of STEM students at this university study between .25 and 4.25 hours the night before an exam.

(b) It is unlikely that a student from this university would study for 6 hours the night before an exam.

(c) The margin of error for our confidence interval was 2 hours.

(d) We are 95% confident that the average number of hours studied by students at this

(e) We are 95% confident that the average number of hours studied by students at this

(f) If we repeated the process of sampling another 100 students and building a 95% confidence interval using this data, there is a 95% chance that the sample average would land between .25 and 4.25 hours.

Carry out a hypothesis test at α = .05 to see if income causes a difference in opinions about the legalization of marijuana. If there is significant evidence construct a 95% confidence interval for the difference.

Question 12) Suppose I want to study the amount of time people spend watching TV on a weekday. I find that the average time is 2.35 hours with a standard deviation of 1.93 hours.

(a) What shape do you expect this distribution to have? Why?

(b) Suppose now I take a sample of size 100. Describe the distribution of the sample mean.

(c) What is the probability that in my sample the average time spent watching TV is between 2 and 2.5 hours?

(d) What is the probability that in my sample the average time spent watching TV is below 1.75 hours?

(e) If you observed the results mentioned in (d) what might you conclude?

Question 13) Suppose that a car company claims their new compact car gets 30 miles per gallon. A consumer advocacy group worries that this new compact car gets less than this. They take a random sample of 15 cars and test their fuel efficiency. They get the following data, in miles per gallon.

(a) Carry out a hypothesis test at α = .05 to see if there is evidence for the advocacy group’s claim. Make sure to confirm all necessary conditions.

(b) Regardless of your answer to (a) build a 95% confidence interval for the average miles per gallon that this type of car gets.

(c) If the advocacy group is worried of falsely accusing the car company of producing cars that get less than 30 miles per gallon should they retest at α = .01 or α = .1?

(a) Build a 90% confidence interval for the true proportion of how often this slot machine pays out. Make sure to use an appropriate method and confirm all necessary conditions.

(b) Using your interval would you consider it be likely, unlikely, or impossible that the true proportion is 2%?

(c) Using your study as a pilot study, how many times would you need to play the machine to build a 90% confidence interval with a margin of error of less than 1%?

Question 15) Suppose you are interested in studying if caffeine decreases reaction time. To do this you take a random sample of 10 people. You bring them to your lab one day and give them a reflex test. The reflex test is done by putting two buttons in front of them, one that says red and one that says blue. A color then flashes on the screen and the subject has to press the correct button. You then bring all the participants back the next day and give them caffeine pills. You wait twenty minutes for the pills to take effect and then give them the test again. You get the following results (note all values are in milliseconds):

(a) Carry out a hypothesis test at α = .01 to test the researcher’s claim. Make sure to confirm all necessary conditions.

(b) Regardless of your answer to (a), build a 99% confidence interval for the difference in reaction time. Make sure to write a sentence interpreting your interval.

(c) Are individual differences in reaction time a flaw in this experiment? Why or why

(d) Give one major flaw in this experiment. How could this be corrected?

Question 16) Suppose you work for a fast food company. You are interested in studying if people spend different amounts when going through the drive-thru versus actually ordering inside the restaurant. You take a random sample of 81 bills from the drive thru and find that the average amount spent was 9.38 with a standard deviation of 3.74. You also take a random sample of 61 bills from dine-in customers and find that the average amount spent was 10.57 with a standard deviation of 4.31.

(a) Carry out a hypothesis test at α = .05 to test this claim. Make sure to confirm all necessary conditions.

(b) Regardless of your answer to (a), construct a 95% confidence interval for the average

(c) Describe what it would mean to make a Type II error in this situation.

Question 17) Suppose you are interested in studying the relationship between college major and whether students enjoy the math classes they take. You take a sample of 150 students from a local university and ask them to identify their major and say if they have enjoyed, felt neutral about, or not enjoyed the math classes they have taken so far. You get the following data:

(a) If you randomly chose someone from your study what is the chance their major was in the social sciences?

(b) If you randomly chose someone from your study what is the chance that their major

(c) If you randomly chose someone from your study what is the chance that their major

(d) If you randomly choose someone from your study what is the chance they enjoyed their math courses given they were in the Arts and Humanities?

(e) Which major had the highest rate of students that felt neutral about the math courses they had taken?

(a) Carry out a hypothesis test at α = .05 to test if there is a relationship between major and enjoyment of math classes. Make sure to confirm all necessary conditions.

Question 18) For this question use the data from Question 18

(b) Describe what it would mean to make a Type I error here.

(c) Describe what it would mean to make a Type II error here.

Question 19) Suppose you work for a large town government. You are studying the ethnic diversity of your area. From previous research in 2012 you had the following ethnic breakdown:

(a) Test at α = .05 to see if the ethnic diversity in your area has changed since 2012. Make sure to check all necessary conditions and state your conclusion in the context of this situation.

(b) If you were worried about missing a change in the ethnic diversity of your area would you retest at α=.01or α=.10? Why?

Question 20) Suppose you are interested in studying the average number of hours students study the night before an exam. You take a random sample of 100 students from a local university all of whom are majoring in one of the STEM fields. After asking them to estimate how many hours they usually study the night before an exam you get the following 95% confidence interval in hours, (.25, 4.25).

Decide if the following interpretations are true or false.

(a) 95% of STEM students at this university study between .25 and 4.25 hours the night before an exam.

(b) It is unlikely that a student from this university would study for 6 hours the night before an exam.

(c) The margin of error for our confidence interval was 2 hours.

(d) We are 95% confident that the average number of hours studied by students at this

(e) We are 95% confident that the average number of hours studied by students at this

(f) If we repeated the process of sampling another 100 students and building a 95% confidence interval using this data, there is a 95% chance that the sample average would land between .25 and 4.25 hours.

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