Say the fraction of students at college 1 who took statistics is given by 1/5. Say the fraction of students at college 2 who took statistics is given by 1/2. You survey 10 people from each college (so 20 people in total).

(a) What is the probability that more than half of the surveyed students took statistics from college 1?

(b) What is the probability that more than half of the surveyed students took statistics from college 2?

(c) Assuming that the colleges are independent, what is the probability that more than half of the surveyed students took statistics from college 1 and more than half of the surveyed students took statistics from college 2?

(d) Assuming that the colleges are independent, what is the probability that more than half of the surveyed students took statistics from college 1 or more than half of the surveyed students took statistics from college 2?

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Now say you redo your survey of college 1, but you have not decided how many students to ask, so you just keep asking students indefinitely.

(a) What is the probability that none of the first 3 people have taken statistics?

(b) What is the probability that the first person you find who takes statistics is the 5th person you ask?

(c) What is the probability that the first person you find who takes statistics is the 5th person you ask, given that none of the first 3 people have taken statistics?

It has been extensively demonstrated that adequate night sleep is crucial for cognitive performance.  To investigate whether first year USC students are getting adequate sleep the night before a final exam, 81 randomly selected first year USC students were asked to record their number of hours of sleep the night before their final exams. The mean number of hours of sleep among the 81 students was 6.2 with an SD of 1.5 hours.                                                                                       (18 points)

1. Identify the population, sample, and parameter of interest in this study.  (3 points)
2. Construct a 95% CI for the mean number of hours first year USC students sleep the night before an exam.  Explain the meaning of this CI (you are expected to explain what this confidence interval says about the parameter of interest)    (5 points)
3. Now construct a 99% CI for the mean number of hours first year USC students sleep the night before an exam.  Explain the meaning of this CI and contrast with the 95% CI in part b.                          (5 points)
4. Suppose the 81 students represent a subset of a larger pool of 100 students randomly selected for the study.  The 81 students provided the investigators with their number of hours of sleep, while the remaining 19 students did not respond when contacted by the investigators.  If the 19 ‘non-responders’ were more likely to sleep more hours before an exam than a typical first year USC student, how would this affect the estimate and CI for the mean number of hours first year USC students sleep the night before an exam?                                                                                                                (5 points)

1. The grades on a chemistry midterm at Springer are roughly symmetric with μ=67 and σ=2.0. William scored 66 on the exam.

1. Find the z-score for William's exam grade. Round to two decimal places.
2. What percentage of students in this midterm scored less than William?
3. What percentage of students scored more than 70 in this midterm?

2. The grades on a language midterm at Oak are roughly symmetric with μ=67 and σ=2.5. Ishaan scored 70 on the exam.

1. Find the z-score for Ishaan's exam grade. Round to two decimal places.
2. What percentage of students in this midterm scored less than Ishaan?
3. What percentage of students scored more than 70 in this midterm?

3. The grades on a math midterm at Springer are roughly symmetric with μ=78 and σ=5.0. Omar scored 70 on the exam.

1. Find the z-score for Omar's exam grade. Round to two decimal places.
2. What percentage of students in this midterm scored less than Omar?
3. What percentage of students scored more than 80 in this midterm?
4. What percentage of students scored between 70 and 90 in this midterm?

4. The grades on a geometry midterm at Springer are roughly symmetric with μ=68 and σ=2.0. Emily scored 69 on the exam.

1. Find the z-score for Emily's exam grade. Round to two decimal places.
2. What percentage of students in this midterm scored less than Emily?
3. What percentage of students scored more than 75 in this midterm?
4. What percentage of students scored between 65 and 75 in this midterm?

Med Student Sleep Average (Raw Data, Software Required):
Here we consider a small study on the sleep habits of med students and non-med students. The study consists of the hours of sleep per night obtained from 30 non-med students and 25 med students. The sample data is given in the table below. Test the claim that, on average, the mean hours of sleep for all med students is different from that for non-med students. Test this claim at the 0.01 significance level.

Med Student Sleep Average (Raw Data, Software Required):
Here we consider a small study on the sleep habits of med students and non-med students. The study consists of the hours of sleep per night obtained from 30 non-med students and 25 med students. The sample data is given in the table below. Test the claim that, on average, the mean hours of sleep for all med students is different from that for non-med students. Test this claim at the 0.05 significance level.

Please give a detailed explanation, not just the response. If possible, a way to work out the problem in Excel is helpful! Thanks in advance!

According to the Carnegie unit system, the recommended number of hours students should study per unit is 2. Are statistics students' study hours less than the recommended number of hours per unit? The data show the results of a survey of 16 statistics students who were asked how many hours per unit they studied. Assume a normal distribution for the population.

0, 3.2, 0, 3.2, 2, 1.3, 0.1, 0, 3.1, 2.1, 2.4, 1.5, 1.5, 2.4, 0.8, 2.6

What can be concluded at the αα = 0.01 level of significance?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:

H0:H0:  ? μ p  Select an answer ≠ < = >       

H1:H1:  ? p μ  Select an answer ≠ > < =    

3. The test statistic ? z t  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer accept reject fail to reject  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population mean is not significantly less than 2 at αα = 0.01, so there is sufficient evidence to conclude that the population mean study time per unit for statistics students is equal to 2.
   • The data suggest the populaton mean is significantly less than 2 at αα = 0.01, so there is sufficient evidence to conclude that the population mean study time per unit for statistics students is less than 2.
   • The data suggest that the population mean study time per unit for statistics students is not significantly less than 2 at αα = 0.01, so there is insufficient evidence to conclude that the population mean study time per unit for statistics students is less than 2.
8. Interpret the p-value in the context of the study.
   • If the population mean study time per unit for statistics students is 2 and if you survey another 16 statistics students, then there would be a 11.81302557% chance that the sample mean for these 16 statistics students would be less than 1.64.
   • There is a 11.81302557% chance of a Type I error.
   • If the population mean study time per unit for statistics students is 2 and if you survey another 16 statistics students, then there would be a 11.81302557% chance that the population mean study time per unit for statistics students would be less than 2
   • • There is a 11.81302557% chance that the population mean study time per unit for statistics students is less than 2.
   • Interpret the level of significance in the context of the study.
     • There is a 1% chance that the population mean study time per unit for statistics students is less than 2.
     • There is a 1% chance that students just don't study at all so there is no point to this survey.
     • If the population mean study time per unit for statistics students is less than 2 and if you survey another 16 statistics students, then there would be a 1% chance that we would end up falsely concluding that the population mean study time per unit for statistics students is equal to 2.
     • If the population mean study time per unit for statistics students is 2 and if you survey another 16 statistics students, then there would be a 1% chance that we would end up falsely concluding that the population mean study time per unit for statistics students is less than 2.

A researcher wants to determine whether high school students who attend an SAT preparation course score significantly different on the SAT than students who do not attend the preparation course. For those who do not attend the course, the population mean is 1050 (μ = 1050). The 16 students who attend the preparation course average 1150 on the SAT, with a sample standard deviationof 300. On the basis of these data, can the researcher conclude that the preparation course has a significant difference on SAT scores? Set alpha equal to .05.Q78:The appropriate statistical procedure for this example would be aA.z-testB.t-testQ79:Is this a one-tailed or a two-tailed test?A.one-tailedB.two-tailed

11Q80:The most appropriate null hypothesis (in words) would beA.There is no statistical difference in SAT scores when comparing students who took theSAT prep course with the general population of students who did not take the SAT prep course.B.There is a statistical difference in SAT scores when comparing students who took the SAT prep course with the general population of students who did not take theSAT prep course.C.The students who took the SAT prep course did not score significantly higher on the SAT when compared to the general population of students who did not take the SAT prep course.D.The students who took the SAT prep course did score significantly higher on the SAT when compared to the general population of students who did not take the SAT prep course.Q81:The most appropriate null hypothesis (in symbols) would beA.μSATprep= 1050B.μSATprep= 1150C.μSATprep1050D.μSATprep1050Q82:Set up the criteria for making a decision. That is, find the critical value using an alpha = .05. (Make sure you are sign specific: + ; -; or ) (Use your tables)Summarize the data into the appropriate test statistic.Steps:Q83:What is the numeric value ofyour standard error?Q84:What is the z-value or t-value you obtained (your test statistic)?Q85:Based on your results (and comparing your Q84 and Q82 answers) would youA.reject the null hypothesisB.fail to reject the null hypothesisQ86:The best conclusion for this example would beA.There is no statistical difference in SAT scores when comparing students who took the SAT prep course with the general population of students who did not take the SAT prep course.B.There is a statistical difference in SAT scores when comparing students who took the SAT prep course with the general population of students who did not take the SAT prep course.C.The students who took the SAT prep course did not score significantly higher on the SAT when compared to the generalpopulation of students who did not take the SAT prep course.D.The students who took the SAT prep course did score significantly higher on the SAT when compared to the general population of students who did not take the SAT prep course.

12Q87:Based on your evaluation of the null in Q85 and your conclusion is Q86, as a researcher you would be more concerned with aA.Type I statistical errorB.Type II statistical errorCalculate the 99%confidence interval.Steps:Q88:The mean you will use for this calculation isA.1050B.1150Q89:What is the new critical value you will use for this calculation?Q90:As you know, two values will be required to complete the following equation:__________ __________Q91:Which of the following is a more accurate interpretation of the confidence interval you just computed?A.We are 99% confident that the scores fall in the interval _____ to _____.B.We are 99% confident that the average score on the SAT by the students who took the prep course falls in the interval _____ to _____.C.We are 99% confident that the example above has correct values.D.We are 99% confident that the differencein SAT scores between the students who took the prep course and the students who did not falls in the interval _____ to ____

A researcher wants to determine whether high school students who attend an SAT preparation course score significantly different on the SAT than students who do not attend the preparation course. For those who do not attend the course, the population mean is 1050 (μ = 1050). The 16 students who attend the preparation course average 1150 on the SAT, with a sample standard deviationof 300. On the basis of these data, can the researcher conclude that the preparation course has a significant difference on SAT scores? Set alpha equal to .05.Q78:The appropriate statistical procedure for this example would be aA.z-testB.t-testQ79:Is this a one-tailed or a two-tailed test?A.one-tailedB.two-tailed 11Q80:The most appropriate null hypothesis (in words) would beA.There is no statistical difference in SAT scores when comparing students who took theSAT prep course with the general population of students who did not take the SAT prep course.B.There is a statistical difference in SAT scores when comparing students who took the SAT prep course with the general population of students who did not take theSAT prep course.C.The students who took the SAT prep course did not score significantly higher on the SAT when compared to the general population of students who did not take the SAT prep course.D.The students who took the SAT prep course did score significantly higher on the SAT when compared to the general population of students who did not take the SAT prep course.Q81:The most appropriate null hypothesis (in symbols) would beA.μSATprep= 1050B.μSATprep= 1150C.μSATprep1050D.μSATprep1050Q82:Set up the criteria for making a decision. That is, find the critical value using an alpha = .05. (Make sure you are sign specific: + ; -; or ) (Use your tables)Summarize the data into the appropriate test statistic.Steps:Q83:What is the numeric value ofyour standard error?Q84:What is the z-value or t-value you obtained (your test statistic)?Q85:Based on your results (and comparing your Q84 and Q82 answers) would youA.reject the null hypothesisB.fail to reject the null hypothesisQ86:The best conclusion for this example would beA.There is no statistical difference in SAT scores when comparing students who took the SAT prep course with the general population of students who did not take the SAT prep course.B.There is a statistical difference in SAT scores when comparing students who took the SAT prep course with the general population of students who did not take the SAT prep course.C.The students who took the SAT prep course did not score significantly higher on the SAT when compared to the generalpopulation of students who did not take the SAT prep course.D.The students who took the SAT prep course did score significantly higher on the SAT when compared to the general population of students who did not take the SAT prep course. 12Q87:Based on your evaluation of the null in Q85 and your conclusion is Q86, as a researcher you would be more concerned with aA.Type I statistical errorB.Type II statistical errorCalculate the 99%confidence interval.Steps:Q88:The mean you will use for this calculation isA.1050B.1150Q89:What is the new critical value you will use for this calculation?Q90:As you know, two values will be required to complete the following equation:__________ __________Q91:Which of the following is a more accurate interpretation of the confidence interval you just computed?A.We are 99% confident that the scores fall in the interval _____ to _____.B.We are 99% confident that the average score on the SAT by the students who took the prep course falls in the interval _____ to _____.C.We are 99% confident that the example above has correct values.D.We are 99% confident that the difference in SAT scores between the students who took the prep course and the students who did not falls in the interval _____ to ____

In a student community, 30% of the students own a car and 50% of the students who own a car also own a bicycle. Also, 60% of the student community own a bicycle. Furthermore 25% of students who own a bicycle, also own a two-wheeler. Car owners do not own two-wheelers. Finally, 30% of the students own a two-wheeler. What is the probability that a randomly selected student (a) owns a bicycle and a two-wheeler? (b) owns a car, but does not own a bicycle? (c) does not own any of the three? (d) owns a bicycle, but no car or two-wheeler?

At this college, 28% are students of color, 6% of students are veterans, and 1.32% are both veterans and students of color.

(a) If a randomly selected student is a student of color, what is the probability that the student is a veteran?

(b) Is the event of being a student of color independent of the event of being a veteran? Explain your answer by using probability concepts.