Math 116 – 05: Final Examination
Fall 2012
Name:
Each of problems 1 – 6 are worth 4 points.
1. Suppose your local school district decides to randomly test high school students for attention deficit disorder (ADD). There are three high schools in the district, each with grades 9-12. The school board pools all of the students together and randomly samples 250 students. Is this a simple random sample?
(a) Yes, because the students were chosen at random.
(b) Yes, because each student is equally likely to be chosen.
(c) No, because we can’t guarantee there are students from each school in the sample.
(d) No, because we can’t guarantee that there are students from each grade in the sample.
2. In an intro stats class, 57% of students eat breakfast in the morning and 80% of students floss their teeth. Forty-six percent of students eat breakfast and floss their teeth. What is the probability that a student from this class eats breakfast but does not floss his/her teeth?
(a) 9% (b) 11% (c) 34% (d) 57% (e) 91%
3. Five juniors and four seniors have applied for two open student council positions. School administrators have decided to pick the two new members randomly. What is the probability that one junior and one senior are chosen for the two positions?
(a) 16.7% (b) 27.8% (c) 44.4% (d) 55.6% (e) none of these
4. A friend of yours plans to toss a fair coin 200 times. You watch the first 40 tosses, noticing that she got only 16 heads. But then you get bored and leave. If the coin is fair, how many heads do you expect her to have when she has finished the 200 tosses?
(a) 80 (b) 92 (c) 96 (d) 100 (e) 116
5. Which is true about a 98% confidence interval for a population proportion based on a given sample?
I. We are 98% confident that the sample proportion is in our interval.
II. We are 98% confident that the population proportion is in our interval.
III. The interval is wider than a 95% confidence interval would be.
(a) I only (b) II only (c) III only (d) II and III (e) I and II
6. In a certain hypothesis test, a P-value of 0.104 is calculated. Which of the following is true?
(a) There is a 10.4% chance that the null hypothesis is true.
(b) There is a 10.4% chance that the alternative hypothesis is true.
(c) There is a 10.4% chance of the observed statistic given the null hypothesis is true.
(d) All of the above
(e) None of the above
7. Suppose we wanted to construct a 98% confidence interval for the average life span of a new battery with a margin of error of no more than 2.5 hours. Suppose, also, that previous studies on this new battery support that the average deviation of lifespans is about 8.4 hours. How many batteries must be tested to accomplish this? (7 points)
8. Suppose we wanted to construct a 95% confidence interval for the proportion of people in a certain city that are pleased with how their local government is representing them with a margin of error of no more than 2 percentage points. How many people should be chosen from this community at random for the study? (7 points)
9. A certain company sells three models of digital cameras (X, Y, and Z). By studying their sales over a long period of time, it was found that of all cameras sold 20% are model X, 30% are model Y, the rest are model Z. When a camera is purchased, an extended warranty is available. Of those that bought model X, 40% purchased the extended warranty. Of those that bought model Y, 45% purchased the extended warranty. Of those that bought model Z, 60% bought an extended warranty. Define event W as “person bought an extended warranty”. Find the following probabilities and explain what each means for a randomly selected customer at this company. (5 points each) HINT: A tree diagram may be helpful.
(a)
(b)
(c)
(d)
10. A company has developed a new light bulb that they claim has a life span 4200 hours with a standard deviation of 72 hours. What is the probability that in a sample of 100 such bulbs, the average life span is less than 4175 hours? (8 points)
11. A company that manufactures computer chips believes that only 2% of their chips are defective. Suppose that this is true.
(a) What is the smallest sized sample for which we could say that the distribution of the variable is normally distributed? (3 points)
(b) What is the probability that in a sample of 1000 of this company’s computer chips, more than 3% are defective? (7 points)
12. A certain state’s Department of Education randomly selected 500 students and found that 58 of them attended private school. Construct a 98.5% confidence interval for the proportion of all students in this state that attend private school. Do not forget to verify that the process used is valid. (6 points)
13. For each of the following hypothesis tests, calculate the p-value and give the appropriate conclusion at the a = 0.05 level. (6 points each)
(a)
(b)
(c)
14. The numbers below give summary data for the average ACT scores for incoming freshman at a certain university for two groups: those receiving football scholarships and those not receiving football scholarships. (5 points each)
/ s / nScholarship / 21.86 / 2.84 / 44
No Scholarship / 24.75 / 3.29 / 31
(a) Construct a 98% confidence interval for the difference in average ACT scores for football scholarship students and non-football scholarship students.
(b) Based on your answer to (a), would you conclude that average ACT scores are the same for these two populations or not. Support your answer.
15. A legislator is curious whether a majority of the voters in her district favor a law that would reduce the legal blood alcohol level that defines “legally drunk”. She has her staff collect data for analysis. They find that of 260 randomly selected voters in the district, 142 would favor such a law. Does this data provide sufficient evidence that a majority (more than 50%) of the voters are in favor of such a law? Test the relevant hypotheses at the a = 0.05 level. (10 points)
H0: Validity:
Ha:
Test Statistic:
P-Value:
Conclusion:
16. A certain car manufacturer is planning to offer its newest model car in three possible colors: silver, blue, and green. The manager of the company believes that 50% of the cars sold will be silver, 30% will be blue and 20% will be green, and so production is planned accordingly. An independent researcher is curious if these proportions are correct, so 200 people planning to buy a new car in the next year are randomly chosen and asked which color they would choose. The results are shown below. Check if the manager’s proportions are accurate by testing the relevant hypotheses at the a = 0.05 level. (10 points)
Color / Silver / Blue / Green# People / 88 / 80 / 32
H0: Validity:
Ha:
Test Statistic:
P-Value:
Conclusion:
17. In a recent study, the subjects were asked, “How many of your friends do you consider close enough to that you would discuss important personal matters with them?” The researcher collected the data, and its summary is below. Does the data support that, on average, females have more “close friends” than men. Test the appropriate hypotheses at the a = 0.05 level.
(10 points)
n / / sMale / 645 / 1.861 / 1.777
Female / 813 / 2.089 / 1.760
H0: Validity:
Ha:
Test Statistic:
P-Value:
Conclusion:
18. A study was conducted to investigate how effective a new nicotine lozenge is at helping people quit smoking. Subjects were monitored for a year. Of the 459 subjects who had been taking the nicotine lozenge, 82 successfully abstained from smoking for the year. Of the 458 subjects who had been taking a placebo lozenge, 44 successfully abstained from smoking. Does this data support that taking the nicotine lozenge increases the chance of quitting smoking (at least for a year)? Test the relevant hypotheses at the a = 0.05 level. (10 points)
H0: Validity:
Ha:
Test Statistic:
P-Value:
Conclusion: