Name Period 0 and alternative hypothesis Ha

Name _______________________________ Testing a Claim Homework

Period _____________

For 1 - 5, state the appropriate null hypothesis H0 and alternative hypothesis Ha in each case. Be sure to define your parameter each time.

1. Simon reads a newspaper report claiming that 12% of all adults in the U.S. are left-handed. He wonders if

12% of the students at his large public high school are left-handed. Simon chooses an SRS of 100

students and records whether each student is right- or left-handed.

Parameter: Proportion of students at Simon's high school that are left-handed.

0: = .12

: .12

2. A Gallup Poll report on a national survey of 1028 teenagers revealed that 72% of teens said they seldom

or never argue with their friends. Yvonne wonders whether this national result would be true in her large

high school. So she surveys a random sample of 150 students at her school.

Parameter: Proportion of students at Yvonne's high school who say that seldom or never argue with

friends.

0: = .72

: .72

3. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students'

attitudes toward school and study habits. Scores range from 0 to 200. The mean score for U.S. college

students is about 115. A teacher suspects that older students have better attitudes toward school. She

gives the SSHA to an SRS of 45 of the over 1000 students at her college who are at least 30 years of

age.

Parameter: Mean score on the SSHA for students at least 30 years old at the teacher's college.

0: = 115

: > 115

4. Hemoglobin is a protein in red blood cells that carries oxygen form the lungs to body tissues. People with

less than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic. A public health official in

Jordan suspects that Jordanian children are at risk of anemia. He measures a random sample of 50

children.

Parameter: Mean grams of hemoglobin per deciliter of blood for Jordanian children.

0: = 12

: < 12

5. During the winter months, the temperatures at the Colorado cabin owned by the Starnes family can stay

will below freezing (32?F) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the

thermostat at 50?F. The manufacturer claims that the thermostat allows variation in home temperature

of = 3?F. Mrs. Starnes suspects that the manufacturer is overstating how well the thermostat works.

Parameter: Standard deviation of the temperature in the Starnes' cabin.

0: = 3

: > 3

In exercises 6-7, explain what's wrong with the stated hypotheses. Then give the correct hypotheses.

6. A change is made that should improve student satisfaction with the parking situation at a local high

school. Right now, 37% of students approve of the parking that's provided. The null hypothesis 0: > .37 is tested against the alternative : = .37. The null hypothesis should be the current situation (always has =) and the alternative should be the claim.

0: = .37

: > .37

7. In planning a study of the birth weights of babies whose mothers did not see a doctor before delivery, a

researcher states the hypotheses as

0: = 1000

: < 1000

Hypothesis should be about population parameters, not sample statistics.

0: = 1000

: < 1000

8. In Simon's SRS from #1, 16 of the students were left-handed. A significance test yields a P-value of 0.2184. a) Interpret this result in context. If the proportion of lefties at Simon's school is really .12, there is a 21.84% chance of finding a sample of 100 people with a value of as far from .12 as the sample value in either direction. b) Do the data provide convincing evidence against the null hypothesis? Explain. No, something that happens over 20% of the time just by chance when H0 is true is not strong evidence against H0.

9. For Yvonne's survey in #2, 96 students in the sample said they rarely or never argue with friends. A significance test yields a P-value of 0.0291. a) Interpret this result in context. If the proportion of students at Yvonne's school that rarely or never argue with friends is really .72, there is a 2.91% chance of finding a sample of 150 students with a value of as far from .72 as the sample value in either direction. b) Do the data provide convincing evidence against the null hypothesis? Explain. Yes, something that happens about 3% of the time just by chance when H0 is true is strong evidence against H0.

10. In the study of older students' attitudes from #3, the sample mean SSHA score was 125.7 and the sample standard deviation was 29.8. A significance test yields a P-value of 0.0101. a) Interpret the P-value in context. If the mean SSHA score for older students at the teacher's college is really 115, there is a 1.01% chance of finding a sample of 45 students with a mean score of at least 125.7. b) What conclusion would you make if = 0.05? If = 0.01? Justify your answer. Since the p-value of .0101 is less than .05 we would Reject H0 at the = 0.05 level. Since the p-value of .0101 is greater than .01 we would Fail to Reject H0 at the = 0.01 level.

11. For the study of Jordanian children in #4, the sample mean hemoglobin level was 11.3 g/dl and the sample standard deviation was 1.6 g/dl. A significance test yields a P-value of 0.0016. a) Interpret the P-value in context. If the mean hemoglobin level for Jordanian children is really 12 g/dl, there is a .16% chance of finding a sample of 50 children with a mean hemoglobin level less than 11.3 g/dl. b) What conclusion would you make if = 0.05? If = 0.01? Justify your answer. Since the p-value of .0016 is less than .05 and .01 we would Reject H0 for both.

12. Explain in plain language why a significance test that is significant at the 1% level must always be significant at the 5% level. If a test is significant at the 5% level, what can you say about its significance at the 1% level? Significance at the = 0.01 level means that the p-value is less than .01 and therefore also has to be less than .05. Significance at the = 0.05 level means that the p-value is less than .05, but may not be less than .01 so therefore we can't make a conclusion at = 0.01.

13. Slow response times by paramedics, firefighters, and policemen can have serious consequences for

accident victims. In the case of life-threatening injuries, victims generally need medical attention within

8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such

city, the mean response time to all accidents involving life-threatening injuries last year was = 6.7

minutes. Emergency personnel arrived within 8 minutes after 78% of all calls involving life-threatening

injuries last year. The city manager shares this information and encourages these first responders to "do

better". At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening

injuries and examines response times.

a) State the hypotheses for a significance test to determine whether the average response time

has decreased. Be sure to define the parameter of interest.

Parameter: Mean response time to all accidents involving life-threatening injuries in the city.

0: = 6.7

: < 6.7

b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. Type I error: The city concludes that the response time has decreased when it really hasn't. Consequences could be they do not devote more resources and people could die. Type II error: The city concludes that the response time has not improved when it really has. Consequences could be that they continue to spend resources to improve response times that have already improved.

c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer. A Type I error is more serious. The city may stop trying to improve response times because they think they have met their goal, when they haven't. More people could die which is more serious than losing money trying to improve response times that have already improved.

14. You are thinking about opening a restaurant and are searching for a good location. From research you

have done, you know that the mean income of those living near the restaurant must be over $85,000 to

support the type of upscale restaurant you wish to open. You decide to take a simple random sample of 50

people living near one potential location. Based on the mean income of this sample, you will decide whether

to open a restaurant there.

a) State appropriate null and alternative hypotheses. Be sure to define your parameter.

Parameter: Mean income of residents near the restaurant.

85000

: > 85000

0: =

b) Describe a Type I and a Type II error, and explain the consequences of each. Type I error: Open the restaurant in a location where the residents will not be able to support it. Consequences could be they go out of business and lose their money. Type II error: Do not open the restaurant in a location where the residents would have been able to support it. Consequences could be that they lose out on an opportunity to make money.

c) If you had to choose one of the "standard" significance levels for your significance test, would you choose = 0.01, 0.05, 0.10? Justify your choice. I would choose = 0.01 because it minimizes the chances of making a Type I error; the costlier of the two.

15. You read that a statistical test at significance level = 0.05 has power 0.78. What are the probabilities of Type I and Type II errors for this test? Type I error is .05; probability of Type I error = Type II error is .22; probability of Type II error is 1 ? power.

16. A drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer's disease will experience nausea. To test this claim, a significance test is carried out of 0: = 0.10 : < 0.10 You learn that the power of this test at the 5% significance level against the alternative p = 0.08 is 0.64. a) Explain in simple language what "power = 0.64" means in this setting. If p = .08, the probability of correctly Rejecting the null hypothesis is .64.

b) You could get higher power against the same alternative with the same by changing the number of measurements you make. Should you make more measurements or fewer to increase power. Explain. More measurements; increasing the sample size increases the power.

c) If you decide to use = 0.01 in place of = 0.05, with no other changes in the test, will the power increase or decrease? Justify your answer. If you shift your interest to the alternative p = 0.07 with no other changes, will the power increase or decrease? Justify your answer. Decrease. Decreasing increases which decreases the power. Increase. Power increases when you move away from the mean.

For 17-18, check that the conditions for carrying out a one-sample z test for the population proportion p are met.

17. Simon reads a newspaper report claiming that 12% of all adults in the U.S. are left-handed. He wonders if 12% of the students at his large public high school are left-handed. Simon chooses an SRS of 100 students and records whether each student is right- or left-handed. RANDOM: stated that an SRS of 100 was used NORMAL: (.12)(100) = 12, (.88)(100) = 88, number of success and failures are both > 10 INDEPENDENT: (10)(100) = 1000; there are more than 1000 students at a large high school

18. You toss a coin 10 times to test the hypothesis 0: = 0.5 that the coin is balanced. RANDOM: tossing a coin can be considered a randomized experiment NORMAL: (.5)(10) = 5, NORMAL CONDITION NOT MET!! INDEPENDENT: Each coin toss is independent and has no impact on the next.

19. Refer to #17. In Simon's SRS, 16 of the students were left-handed.

a) Calculate the test statistic.

16 = 100 = .16

. 16 - .12

=

= 1.23091

(.

12)(. 88) 100

b) Find the P-value using Table A. Show this result as an area under a standard Normal curve. z = 1.23 p-value is .8907 1 - .8907 = .1093 2(.1093) = .2186

20. A Gallup Poll report on a national survey of 1028 teenagers revealed that 72% of teens said they seldom

or never argue with their friends. Yvonne wonders whether this national result would be true in her large

high school. So she surveys a random sample of 150 students at her school and finds that 96 students

said they seldom or never argue with friends.

a) Calculate the test statistic.

96 = 150 = .64

. 64 - .72

=

= -2.18218

(.

72)(. 28) 150

b) Find the P-value using Table A. Show this result as an area under a standard Normal curve. z = -2.18 p-value is .0146 2(.0146) = .0292

21. A test of 0: = 0.5 versus : > 0.5 has test statistic z = 2.35. a) What conclusion would you draw at the 5% significance level? At the 1% level? P(z > 2.35) = 1 - .9906 = .0094 Reject null at 5%, Reject null at 1% b) If the alternative hypothesis was : 0.5, what conclusion would you draw at the 5% significance level? At the 1% level? 2(.0094) = .0188 Reject null at 5%, Fail to Reject null at 1%

22. A local high school makes a change that should improve student satisfaction with the parking situation.

Before the change, 37% of the school's students approved of the parking that was provided. After the

change, the principal surveys an SRS of 200 of the over 2500 students at the school. In all, 83 students

say that they approve of the new parking arrangement. The principal cites this as evidence that the

change was effective. Perform a test of the principal's claim at the = 0.05 significance level.

Parameter: proportion of students who approve of the new parking arrangement.

0: = .37

: > .37

= .05

1 sample z test for proportions if conditions are met

RANDOM: stated an SRS of 200 students was used

NORMAL: 83 successes and 117 failures are both > 10

INDEPENDENT: 10(200) = 2000, there are more than 2000 students at this school

All conditions met 83

= 200 = .415

. 415 - .37

=

= 1.31812

(.

37)(. 63) 200

( > 1.32) = 1 - .9066 = .0934

P-value = .0934

Since the p-value of .0934 > .05 we Fail to Reject the null hypothesis. There is not sufficient evidence

to support the principal's claim that the change increased student satisfaction with the parking

situation.

23. Refer to #22. a) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. Type I error: Conclude that more that 37% were satisfied when it was only 37%. Consequences are that no further actions are taken and students remain unsatisfied. Type II error: Conclude that only 37% are satisfied when, in fact, more than 37% are satisfied. Consequences are that they continue to take action although no further action is necessary. b) The test has a power of 0.75 to detect that p = 0.45. Explain what this means. If p really is .45, the probability of correctly Rejecting the null is .75. c) Identify two ways to increase the power in part (b). Increase the sample size or increase the significance level ().

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