Name:_________________________



Name:_________________________

Date:__________Period:_________

Test of Significance TEST

Part 1: Multiple Choice. Hand write the letter corresponding to the best answer in space provided on page 7.

(2 points each)

_____1. An automobile manufacturer claims that the average gas mileage of a new model is 35 miles per gallon (mpg). A consumer group is skeptical of this claim and thinks the manufacturer may be overstating the average gas mileage. If [pic] represents the true average gas mileage for this new model, which of the following gives the null and alternative hypotheses that the consumer group should test?

a) Ho: [pic] < 35 mpg

Ha: [pic] 35 mpg

b) Ho: [pic] 35 mpg

Ha: [pic] > 35 mpg

c) Ho: [pic] = 35 mpg

Ha: [pic] > 35 mpg

d) Ho: [pic] = 35 mpg

Ha: [pic] < 35 mpg

e) Ho: [pic] = 35 mpg

Ha: [pic] 35 mpg

_____2. In a test of the null hypothesis Ho: [pic] = 10 against the alternative hypothesis Ha: [pic] > 10, a sample from a normal population produces a mean of 13.4. The z-score for the sample is 2.12 and the p-value is 0.017. Based on these statistics, which of the following conclusions could be drawn?

a) There is reason to conclude that [pic] > 10

b) Due to random fluctuation, 48.3 percent of the time a sample produces a mean larger than 10.

c) 1.7 percent of the time, rejecting the alternative hypothesis is in error.

d) 1.7 percent of the time, the mean is above 10.

e) 98.3 percent of the time, the mean is below 10.

_____3. The process of producing pain-reliever tablets yields tablets with varying amounts of the active ingredient. It is claimed that the average amount of active ingredient per tablet is at least 200 milligrams. The Consumer Watchdog Bureau tests a random sample of 70 tablets. The mean content of the active ingredient for the sample is 194.3 milligrams, while the standard deviation is 21 milligrams. What is the approximate p-value for the appropriate test?

a) 0.012

b) 0.024

c) 0.050

d) 0.100

e) 0.488

_____4. Which of the following is a criterion for choosing a t-test rather than a z-test when making an inference about the mean of a population?

a) The standard deviation of the population is unknown.

b) The mean of the population is unknown.

c) The sample may not have been a simple random sample.

d) The population is not normally distributed.

e) The sample size is less than 100.

_____5. On their birthdays, employees at a large company are permitted to take a 60-minute lunch break instead of the usual 30 minutes. Data were obtained from 10 randomly selected company employees on the amount of time that each actually took for lunch on his or her birthday. The company whishes to investigate whether these data provide convincing evidence that the mean time is greater than 60 minutes. Of the following, which information would NOT be expected to be a part of the process of correctly conducting a hypothesis test to investigate the question, at 0.05 level of significance?

a) Being willing to assume that the distribution of actual birthday lunch times for all employees at the company is approximately normal.

b) Knowing that there are no outliers in the data as indicated by the normal probability plot and the boxplot.

c) Using a t-statistic to carry out the test.

d) Using 9 for the number of degrees of freedom

e) Given that the p-value is greater than 0.05, rejecting the null hypothesis and concluding that the mean time was not greater than 60 minutes.

_____6. A significance test gives a P-value of 0.04. From this we can

a) Reject Ho at the 1% significance level

b) Reject Ho at the 5% significance level

c) Say that the probability that Ho is false is 0.04

d) Say that the probability that Ho is true is 0.04

e) None of the above.

_____7. A certain population follows a normal distribution with mean [pic] and standard deviation [pic] = 2.5. You collect data and test the hypothesis

Ho: [pic] = 1 vs. Ha: [pic] 1

You obtain a P-value of 0.022. Which of the following is true?

a) A 95% confidence interval for [pic] will include the value 1.

b) A 95% confidence interval for [pic] will include the value 0.

c) A 99% confidence interval for [pic] will include the value 1.

d) A 99% confidence interval for [pic] will include the value 0.

e) None of these is necessarily true.

_____8. A significance test was preformed to test the null hypothesis Ho: [pic] = 2 versus the alternative Ha: [pic] 2. The test statistic is z = 1.40. The P-value for this test is approximately

a) 0.16

b) 0.08

c) 0.003

d) 0.92

e) 0.70

f) None of the above.

_____9. A 95% confidence interval for [pic] is calculated to be (1.7, 3.5). It is now decided to test the hypothesis Ho: [pic] = 0 vs. Ha: [pic] 0 at the [pic]= 0.05 level, using the same data as was used to construct the confidence interval

a) We cannot test the hypothesis without the original data.

b) We cannot test the hypothesis at the [pic]= 0.05 level since the [pic]= 0.05 test is connected to the 97.5% confidence interval.

c) We can only make the connection between hypothesis tests and confidence intervals if the sample sizes are larger.

d) We would reject Ho at level [pic]= 0.05.

e) We would accept Ho at level [pic]= 0.05.

_____10. Which of the following are true statements?

I. The alternative hypothesis is stated in terms of a sample statistic.

II. A large P-value indicates strong evidence against the null hypothesis.

III. If a sample is large enough, the necessity for it to be a simple random sample is diminished.

a) I only

b) II only

c) III only

d) Exactly two of the above statements are true.

e) None of the above statements are true.

_____11. In an effort to curb certain diseases, especially autoimmune (AIDS), San Francisco has a program whereby drug users can exchange used needles for fresh ones. As reported in the Journal of the American Medical Association (January 12, 1994, p. 115), 35% of 5644 intravenous drug users in San Francisco admitted to sharing needles. Is this sufficient evidence to say that the rate of sharing needles has dropped from the pre-needle exchange rate of 66%?

a) P < 0.001, so this is very strong evidence that the rate has dropped.

b) P = 0.0063, so this is strong evidence of a drop in rate.

c) P is between 0.01 and 0.05, so there is moderate evidence of a drop in rate.

d) P is between 0.05 and 0.10, so this is some evidence of a drop in rate.

e) P = 0.31, so there is no real evidence of a drop in rate.

_____12. An IRS representative claims that the average deduction for medical care is $1250. A taxpayer who believes that the real figure is lower samples 12 families and comes up with a mean of $934 and a standard deviation of $616. Where is the P-value?

a) Below 0.01

b) Between 0.01 and 0.025

c) Between 0.025 and 0.05

d) Between 0.05 and 0.10

e) Above 0.10

_____13. An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment with major defects prior to using the potatoes to make potato chips. Unless there is clear evidence that this proportion is less than 0.10, she will reject the shipment. To reach a decision she will test the hypotheses

H0: p = 0.10, Ha: p < 0.10

Using a large sample test for a population proportion. To do so, she selects an SRS of 50 potatoes from the over 2000 potatoes on the truck. Suppose that only two of the potatoes sampled are found to have major defects

Which of the following conditions for inference about a proportion using a hypothesis test are violated?

a) The data are an SRS from the population of interest

b) The population is at least 10 times as large as the sample

c) n is so large that both np0 and n(1-p0) are 10 or more, where p0 is the proportion with major defects if the null hypothesis is true

d) There appears to be no violations

e) More than one condition is violated

_____14. In a test of Ho: [pic] = 100 against Ha: [pic] 100. a sample of size 80 produces z = 0.8 for the value of the statistic. The P-value of the test is thus equal to:

a) 0.20

b) 0.40

c) 0.29

d) 0.42

e) 0.21

_____15. A sociologist is studying the effect of having children within the first two years of marriage on the divorce rate. Using hospital birth records, she selects a random sample of 200 couples who had a child within the first two years of marriage. Following up on these couples. She finds that 80 couples are divorced within five years. The current divorce rate is 50% - sadly.

To determine if having children within the first two years of marriage increases the divorce rate we should test

a) Hypotheses H0: p = 0.50, Ha: p [pic]0.50

b) Hypotheses H0: p = 0.50, Ha: p > 0.50

c) Hypotheses H0: p = 0.50, Ha: p < 0.50

d) Hypotheses H0: p = 0.40, Ha: p > 0.50

e) None of the above

Review Questions

_____16. A manufacturer makes light bulbs and claims that their reliability is 98 percent. Reliability is defined to be the proportion of non-defective items that are produced over the long term. If the company’s claim is correct, what is the expected number of non-defective light bulbs in a random sample of 1,000 samples?

a) 20

b) 200

c) 960

d) 980

e) 1,000

_____17. Which of the following can be used to show a cause-and-effect relationship between two variables?

a) A census

b) A controlled experiment

c) An observational study

d) A sample survey

e) A cross-sectional survey

_____18. The Physicians’ Health Study, a large medical experiment involving 22,000 male physicians, attempted to determine whether aspirin could help prevent heart attack. In this study, one group of about 11,000 physicians took aspirin every other day, while a control group took a placebo. After several years, it was determined that the physicians in the group that took aspirin had significantly fewer heart attacks than the physicians in the control group. Which of the following statements explains why it would not be appropriate to say that everyone should take aspirin every other day?

I. The study included only physicians, and different results may occur in individuals in other occupations.

II. The study included only males and there may be different results for females.

III. Although taking aspirin may be helpful in preventing heart attacks, it may be harmful to some other aspects of health.

a) I only

b) II only

c) III only

d) II and III only

e) I, II and III

_____19. A data of test scores is being transformed by applying the following rule to each of the raw scores:

Transformed score = 3.5(raw score) + 6.2

Which of the following is not true?

a) The mean transformed score equals 3.5(the mean raw score) + 6.2

b) The median transformed score equals 3.5(the median raw score) + 6.2

c) The range of the transformed scores equals 3.5(the range of the raw scores) + 6.2

d) The standard deviation of the transformed scores equals 3.5(the standard deviation of the raw scores)

e) The IQR of the transformed scores equals 3.5(the IQR of the raw scores)

_____20. The back-to-back stem-and-leaf plot below gives the percentage of students who dropped out of school at each of the 49 high schools in a large metropolitan school district

|School Year | |School Year |

|1989-1990 | |1992-1993 |

| |0 |4 |

|9 9 9 9 8 8 7 |0 |5 6 6 6 7 7 7 8 8 8 9 |

|4 4 4 4 4 3 3 2 2 2 2 1 1 1 1 0 |1 |0 0 0 0 1 1 1 1 2 2 2 3 3 4 4 4 4 |

|9 9 9 7 7 6 6 6 6 5 |1 |5 5 5 6 6 6 6 7 7 7 7 8 |

|4 2 2 2 1 0 0 |2 |1 3 |

|8 8 8 7 6 |2 | |

|2 |3 |0 1 1 2 |

|7 6 6 |3 |5 |

| |4 | |

For 1989-1990, [pic]represents 32%

For 1992-1993, [pic]represents 12%

Which of the following statements is NOT justified by these data?

a) The drop-out rate decreased in each of the 49 high schools between the 1989-1990 and 1992-1993 school years

b) For the school years shown, most students in the 49 high schools did not drop out of high school.

c) In general, drop-out rates decreased between the 1989-1990 and 1992-1993 school years

d) The median drop-out rate of the 49 high schools decreased between the 1989-1990 and 1992-1993 school years

e) The spread between the schools with the lowest drop out rates and those with the highest drop-out rates did not change much between the 1989-1990 and 1992-1993 school years

Name:___________________________

Date:_____________Period:________

PART I answers (2 points each)

|1. |2. |3. |4. |5. |

|6. |7. |8. |9 |10. |

|11. |12. |13. |14. |15. |

|16. |17. |18. |19. |20. |

PART II - Answer completely, but be concise. Write sequentially and show all steps. Show all your work. Indicate clearly the methods you used, because you will be graded on the correctness of your methods as well as on the accuracy of your results and explanations.

YOU MUST WRITE ANSWERS IN BY HAND – you need not print out the first 6 pages.

21. Suppose that a study has been published linking arsenic to increased cancer rates. (Arsenic is commonly found in tap water.) The study also stated that arsenic levels below 10 ppm could be considered harmless. The residents of Newark, NJ, are worried about the arsenic level in the local tap water and would like to conduct a statistical test to address the question of arsenic levels in their tap water.

a) What are the null and alternate hypotheses they should use? In a few sentences,

justify your choice of the alternate hypothesis (4 points)

b) Describe how you as a researcher hired by the city of Newark would design this study. (Note: You are not asked to actually perform an hypothesis test here – review of chapter 5) (6 points)

22. Since Hill Valley High School eliminated the use of bells between classes, teachers have noticed that more students seem to be arriving at class a few minutes late. One teacher decided to collect data top determine whether the students’ and teachers’ watches are displaying the correct time. At exactly 12 noon, the teacher asked 9 randomly selected students and 9 randomly selected teachers to record the times on their watches to the nearest half-minute. The ordered data showing minutes after 12:00 as positive values and minutes before 12:00 as negative values are shown in the table below (12 points)

Students |-4.5 |-3.0 |-0.5 |0 |0 |0.5 |0.5 |1.5 |5.0 | |Teachers |-2.0 |-1.5 |-1.5 |-1.0 |-1.0 |-0.5 |0 |0 |0.5 | |

a) Construct parallel boxplots using these data

b) Based on the boxplots in part (a), which of the two groups, students or teachers, tends to have watch times that are closer to true time? Explain your choice

c) The teacher wants to know whether individual student’s watches tend to be set correctly. She proposes to test H0: [pic]= 0 versus Ha: [pic][pic] 0 [pic]where [pic] represents the mean amount by which all student’s watches differ from the correct time. Is this an appropriate pair of hypotheses to test to answer the teacher’s question? Explain why or why not. Do not carry out the test.

23. A large University provides housing for 10% of its graduate students to live on campus. The University’s housing office thinks that the percentage of graduate students looking for housing on the campus may be more than 10%. The housing office decides to survey a random sample of graduate students, and 62 of the 481 respondents say they are looking for housing on campus. (16 points)

a) On the basis of the survey data, would you recommend that housing office consider increasing the amount of housing on campus available to graduate students? Give appropriate evidence to support your recommendation

b) In addition to the 481 graduate students who responded to the survey, there were 19 who did not respond. If these 19 had responded, is it possible that your recommendation would have changed? Explain.

24. An important part of any dispensing process is statistical quality control. At the Billy Goat Gruff Inn, machines are set to dispense 600 ml of soda into every customer's glass. Over time, however, the machine can get “out of control” and dispense too much soda or too little. At a random point in time each clock hour, the owner dispenses and checks a glass of dispensed soda and determines the actual volume of soda dispensed. One day the volumes of the dispensed soda were: (12 points)

600.15, 599.92, 599.85, 599.92, 599.81, 600.14, 600.04, 599.98

Is there sufficient evidence to conclude that the dispensing machine needs some adjustment?

25. A pharmaceutical company has developed a new drug to reduce cholesterol. A regulatory agency will recommend the new drug for use if there is convincing evidence that the mean reduction in cholesterol level after one month of use is more than 20 milligrams/deciliter (mg/dl), because a mean reduction of this magnitude would be greater than the mean reduction for the current most widely used drug.

The pharmaceutical company collected data by giving the new drug to a random sample of 50 people from the population of people with high cholesterol. The reduction in cholesterol level after one month of use was recorded for each individual in the sample, resulting in a sample mean reduction and standard deviation of 24 mg/dl and 15mg/dl respectively. (16 points)

a) The regulatory agency decides to use an interval estimate for the population mean reduction in cholesterol level for the new drug. Provide this 95% confidence level. Be sure to interpret this interval

b) Because the 95% confidence interval includes 20, the regulatory agency is not convinced that he new drug is better than the current best-seller. The pharmaceutical company tested the following hypotheses:

H0: [pic]= 20 versus Ha: [pic] > 20

Where [pic] represents the population mean reduction in cholesterol level for the new drug.

The test procedure resulted in a t-value of 1.89 and a p-value of 0.033. Because the p-value was less than 0.05, the company believes that there is convincing evidence that the mean reduction in the cholesterol level for the new drug is more than 20. Explain why the confidence interval and the hypothesis test led to different conclusions.

c) The company would like to determine a value L that would allow them to make the following statement:

We are 95% confident that the true mean reduction in cholesterol level is greater than L

A statement of this form is called a one-sided confidence interval. The value of L can be found using the following formula:

L = [pic]

This has the same form as the lower endpoint of the confidence interval in part (A), but requires a different critical value, t*. What value should be used for t* ?

Recall that he sample mean reduction in cholesterol level and the standard deviation are 24mg/dl and 15mg/dl respectively. Compute the value of L.

d) If the regulatory agency had used a one-sided confidence interval in part (c) rather than the interval constructed in part (a), would it have reached a different conclusion?

I pledge that the answers to the questions on this test have been formulated by myself and that I can explain and reproduce all the answers on my own if asked:_______________________________

108 points total/100

108 - ______=______

100 .

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