Test 11A



Significance Test Take Home Exam

Name: AP Statistics

Directions: Work on these sheets.

Part 1: Multiple Choice. Circle the letter corresponding to the best answer.

1. In preparing to use a t procedure, suppose we were not sure if the population was normal. In which of the following circumstances would we not be safe using a t procedure?

(a) A stemplot of the data is roughly bell shaped.

(b) A histogram of the data shows moderate skewness.

(c) A stemplot of the data has a large outlier.

(d) The sample standard deviation is large.

(e) The t procedures are robust, so it is always safe.

2. The heights (in inches) of males in the United States are believed to be normally distributed with mean µ. The average height of a random sample of 25 American adult males is found to be [pic]= 69.72 inches and the standard deviation of the 25 heights is found to be s = 4.15. The standard error of [pic] is

(a) 0.17

(b) 0.69

(c) 0.83

(d) 1.856

(e) 2.04

3. In an opinion poll, 25% of a random sample of 200 people said that they were strongly opposed to having a state lottery. The standard error of the sample proportion is approximately

a) 0.03

b) 0.25

c) 0.0094

d) 6.12

e) 0.06

(f) None of the above.

4. The water diet requires one to drink two cups of water every half hour from when one gets up until one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are

Person 1 2 3 4__

Weight before the diet 180 125 240 150

Weight after six weeks 170 130 215 152

For the population of all adults, assume that the weight loss after six weeks on the diet (weight before beginning the diet – weight after six weeks on the diet) is normally distributed with mean difference[pic]. To determine if the diet leads to weight loss, we test the hypotheses

H0: [pic] = 0, Ha: [pic] > 0.

Based on these data we conclude that

(a) We would not reject H0 at significance level 0.10.

(b) We would reject H0 at significance level 0.10 but not at 0.05.

(c) We would reject H0 at significance level 0.05 but not at 0.01.

(d) We would reject H0 at significance level 0.01.

(e) The sample size is too small to allow use of the t procedures.

5. An opinion poll asks a random sample of adults whether they favor banning ownership of handguns by private citizens. A commentator believes that more than half of all adults favor such a ban. The null and alternative hypotheses you would use to test this claim are:

(a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

(e) None of the above.

The next two questions refer to the following situation: In some mining operations, a byproduct of the processing is mildly radioactive. Of prime concern is the possibility that release of these byproducts into the environment may contaminate the freshwater supply. There are strict regulations for the maximum allowable radioactivity in supplies of drinking water, namely an average of 5 picocuries per liter (pCi/L) or less. However, it is well known that even safe water has occasional hot spots that eventually get diluted, so samples of water are assumed safe unless there is evidence to the contrary. A random sample of 25 specimens of water from a city’s water supply gave a mean of 5.39 pCi/L and a standard deviation of 0.87 pCi/L.

6. The appropriate null and alternative hypotheses are:

a) H0: µ = 5.39 vs Ha: µ (5.39

b) H0: µ = 5.39 vs Ha: µ < 5.00

c) H0: µ = 5 vs Ha: µ = 5.39

d) H0: µ = 5 vs Ha: µ < 5

e) H0: µ = 5 vs Ha: µ > 5

7. The value of the test statistic, the P-value (computed by a calculator), and the decision to reject or fail to reject [pic] at [pic]level are:

(a) z* = 2.24; P-value = 0.0125; reject [pic]

(b) z* = 2.24; P-value = 0.0125; fail to reject [pic]

c) t* = 2.24 with 25 df; reject if t* > 1.708; P-value = 0.0171 reject [pic]

d) t* = 2.24 with 24 df; P-value = 0.0173; reject [pic]

e) t* = 2.24 with 24 df; P-value = 0.0173; fail to reject [pic]

8. In a test of H0: p = 0.4 against Ha: p [pic] 0.4, a sample of size 100 produces z = 1.28 for the value of the test statistic. Thus the P-value (or observed level of significance) of the test is approximately equal to:

a) 0.90

b) 0.40

c) 0.05

d) 0.20

e) 0.10

9. A significance test allows you to reject a hypothesis H0 in favor of an alternative Ha at the 5% level of significance. What can you say about significance at the 1% level?

(a) H0 can be rejected at the 1% level of significance.

(b) There is insufficient evidence to reject H0 at the 1% level of significance.

(c) There is sufficient evidence to accept H0 at the 1% level of significance.

(d) Ha can be rejected at the 1% level of significance.

(e) The answer can’t be determined from the information given.

10. You are testing a [pic]against [pic]based on an SRS of 20 observations from a Normal population. The t statistic is t = -2.25. The P-value

(a) falls between 0.01 and 0.02

(b) falls between 0.02 and 0.04

(c) falls between 0.04 and 0.05

(d) falls between 0.05 and 0.25

(e) is greater than 0.25

11. A 95% confidence interval for a population mean [pic]is calculated to be (1.7, 3.5). Assume that the conditions are met to perform a test of [pic]versus [pic] at the [pic]level based on the confidence interval?

(a) None. We cannot carry out the test without the original data.

(b) None. We cannot draw a conclusion at the [pic]since this test is connected to the 97.5% confidence interval.

(c) None. Confidence intervals and significance tests are unrelated procedures.

(d) We would reject [pic]at level [pic].

(e) We would fail to reject [pic]at level [pic].

12. An SRS of 100 postal employees found that the average time these employees had worked at the postal service was 7 years with a standard deviation of 2 years. Do these data provide convincing evidence that the mean time of employment[pic]for the population of postal employees has changed from the value of 7.5 that was true 20 years ago? To determine this, we test the hypotheses [pic]versus [pic]using a one-sample t test. What conclusion should you draw at the 5% significance level.

(a) There is convincing evidence that the mean time working with the postal service has changed.

(b) There is not convincing evidence that the mean time working with the postal service has changed.

(c) There is convincing evidence that the mean time working with the postal service is still 7.5 years

(d) There is convincing evidence that the mean time working with the postal service is now 7 years.

(e) We cannot draw a conclusion at the 5% significance level. The sample size is too small.

Part 2: Free Response

Answer completely, but be concise. Write sequentially and show all steps.

1. A software company is trying to decide whether to produce an upgrade of one of its programs. Customers would have to pay $100 for the upgrade. For the upgrade to be profitable, the company needs to sell it to more than 20% of their customers. You contact a random sample of 60 customers and find that 16 would be willing to pay $100 for the upgrade.

(a) Do the sample data give good evidence that more than 20% of the company’s customers are willing to purchase the upgrade? Carry out an appropriate test at the [pic]significance level.

(b) Describe a Type I and a Type II error in the context of this problem. Which would be the more serious mistake in this setting and why?

2. A government report says that the average amount of money spent per U.S. household per week on food is about $158. A random sample of 50 households in a sample of 50 households in a small city is selected, and their weekly spending on food is recorded. The Minitab output below shows the results of requesting a confidence interval for the population mean [pic]. An examination of the data reveals no outliers

One-Sample T

N MEAN STDEV SEMEAN 95% CI

50 165.00 20.00 2.83 (159.32, 170.681)

a) Explain why the Normal condition is met in this case.

b) Can you conclude that the mean weekly spending on food in this city differs from the national figure of $158? Give appropriate evidence to support your answer

3. “I can’t get through my day without coffee” is a common statement from many students. Assumed benefits include keeping students awake during lectures and making them mote alert for exams and tests. Students in a statistics class designed an experiment to measure memory retention with and without drinking a cup of coffee one hour before a test. This experiment took place on two different days in the same week (Monday and Wednesday). Ten students were used. Each student received no coffee or one cup of coffee, one hour before the test on a particular day. The test consisted of a series of words flashing on a screen, after which the student had to write down as many of the words as possible. On the other day, each student received a different amount of coffee (none or one cup).

(a) The data from the experiment are provided in the table below. Set up and carry out an appropriate test to determine whether ther is convincing evidence that drinking coffee improves memory.

|Student |No cup |One cup |

|1 |24 |25 |

|2 |30 |31 |

|3 |22 |23 |

|4 |24 |24 |

|5 |26 |27 |

|6 |23 |25 |

|7 |26 |28 |

|8 |20 |20 |

|9 |27 |27 |

|10 |28 |30 |

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