STATISTICS: Sect



AP STATS Name ______________________________

Confidence Intervals (with proportions)

1. [pic]is called the …………………… …………………… and is a …………………… ……………………

Which is a parameter? ……… Which is a statistic? ………

2. A major concern today is the safety of people talking on cell phones while driving. A survey of 12,000 teenagers in Texas found that 5,762 admitted to talking on a cell phone while driving. While this sample is not an SRS, it is close enough that our method gives an approximately correct confidence interval.

a. Find[pic]. ……………

b. What is the population of interest? ………………………………………………………………………………

c. What type of bias is likely to occur in this survey? Do you think the proportion of teenagers talking on a cell phone while driving is higher or lower than stated above? Explain.

……………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………

d. Using the formula for a confidence interval, [pic], find each of the following intervals. Show substitutions into the formula.

90% confidence interval:

95% confidence interval:

99% confidence interval:

e. Write an interpretation of your 90% confidence interval.

………………………………………………………………………………………………………………………

………………………………………………………………………………………………………………………

f. Write an interpretation of your 90% confidence level.

………………………………………………………………………………………………………………………

………………………………………………………………………………………………………………………

3. As the level of confidence increases, the interval length ……………………… (narrows/widens/stays the same)

4. As the sample size increases, the interval length ……………………… (narrows/widens/stays the same)

5. A sample survey found that 79% of 9,132 adults have a landline phone.

a. Find a 90% confidence interval for the true proportion of adults who have a landline phone.

…………………………

d. A 95% confidence interval for the survey is found to be (78.2%, 79.8%). Consider the following statements.

(1) We are 95% certain that between 78.2% and 79.8% of those surveyed have a landline phone.

(2) We are 95% certain that between 78.2% and 79.8% of all adults have a landline phone.

(3) If we repeatedly took more samples of 9,132 adults, then we are 95% certain that 79% will be contained in new intervals.

(4) If we repeatedly took more samples of 9,132 adults, then the true population proportion would be captured in the given interval 95% of the time.

(5) If we repeatedly took more samples of 9,132 adults, then 95% of all intervals created would capture the true population proportion.

Which of these statements is an appropriate interpretation of this interval? ………

Which of these statements is an appropriate interpretation of the level? ………

6. When looking for the sample size for a given margin of error, use ……… for the sample proportion.

7. When looking for the sample size for a given margin of error, how should the answer be rounded? …………

8. Suppose we wish to do a random sample survey with a margin of error of ± 2% at the 90% confidence level. How many people do the pollsters need to interview? Set up an equation and solve.

n = …………

9. Suppose we wish to do a random sample survey with a margin of error of ± 3.5% at the 99% confidence level. How many people do the pollsters need to interview? Set up an equation and solve.

n = …………

10. Suppose we wish to do a random sample survey with a margin of error of ± 4% at the 95% confidence level. How many people do the pollsters need to interview? Set up an equation and solve.

n = …………

STA 2e: Sect. 9.2 Worksheet #1 Name ______________________________

Writing Hypotheses

1. The claim being tested in a significance test is the ………………… hypothesis.

(null/alternative)

2. A test of significance is designed to assess the strength of evidence against the ………………… hypothesis.

(null/alternative)

3. A test of significance is trying to find evidence for the ………………… hypothesis.

(null/alternative)

4. Hypotheses for significance tests must always be expressed in terms of the ……………… (parameter/statistic)

5. Which of these could be possible hypotheses for a significance test?

A H0: = 0.5 B H0: p < 0.5 C H0: p = 0.5 D H0: p = 0.5

Ha: > 0.5 Ha: p ≥ 0.5 Ha: > 0.5 Ha: p > 0.5

6. Which of these could be possible hypotheses for a significance test?

A H0: p > 0.3 B H0: p = 0.3 C H0: = 0.3 D H0: p ≠ 0.3

Ha: p < 0.3 Ha: p ≠ 0.3 Ha: > 0.3 Ha: p = 0.3

For #7 – 10, write only the hypotheses for each problem.

7. According to M&M/Mars Company, 20% of all Plain M&M’s produced are orange. You buy one large 56 oz bag of Plain M&M’s and find that it contains 355 orange M&M’s out of a total of 1,858 M&Ms. Do we have evidence that the company is lying about the proportion of orange M&M’s?

H0: …………………Ha: …………………

8. You’ve been using the Random Digit Table all year, but one day you decide to check out its randomness. You count the number of times the digit “9" occurs in the last four rows of the table and find that there only 12 “9"'s in the last 160 digits. You suspect that the table is flawed by having fewer “9"’s then it should and decide to do a significance test.

H0: …………………Ha: …………………

9. The White House press secretary comments that the president currently has a 72% favorable job approval rating. A pollster challenges this claim as being too high. His polling service has just conducted a random survey of 1000 people (calling both landline and cell phone numbers) and 660 people gave the president a favorable job approval rating. Do we have reason to doubt the press secretary?

H0: …………………Ha: …………………

10. A 2009 nationwide random survey of 1500 adults asked the open-ended question, “What do you think is the most important problem facing this country today?” Sixty-nine percent responded with some form of economic problems (such as economy in general, unemployment/jobs, etc). Do these data provide good evidence that more than 2/3 of all adults believe that economic problems is the most important problem facing this country today?

H0: …………………Ha: …………………

STA 2e: Sect. 9.2 Worksheet #1 p.2

11. If a die is fair, then the number 5 should occur 1/6 of the time. You have a die that you suspect is loaded so that the number 5 lands face up more often then expected. You roll the die 200 times and get 44 5's. Do we have evidence that the die is unfair (i.e. loaded)?

a. What are the mean and standard deviation of the sampling proportion of 5's on the roll of a die?

b. For the sampling distribution at the right, mark the mean of the sampling

proportion, the mean ± 1 standard deviation, mean ± 2 standard deviations,

and the mean ± 3 standard deviations.

c. What is the value of[pic]? …………… Mark this value on the horizontal axis of your sketch. Shade the area to the right of this value.

d. Write the hypotheses for a significance test. H0: …………… and Ha: ……………

e. Using the 68-95-99.7 Rule and the placement of on the graph above, approximately how often would we can a sample proportion as low or lower than ours if the true proportion of 5's really is 1/6?

…………… So is getting 44 5's or more in 200 rolls of the die usual or unusual? ………………

12. According to the 2000 U.S. Census, 80.4% of all U. S. residents 25 years old or older have at least a high school diploma. In 2005, a random sample of 3,000 residents 25 years old or older found that 2,365 had at least a high school diploma. Is there is evidence to suggest that the number of high school graduates is lower than reported in the Census?

a. What are the mean and standard deviation of the sampling proportion?

b. For the sampling distribution at the right, mark the mean of the sampling proportion, the mean ± 1 standard deviation, mean ± 2 standard deviations, and the mean ± 3 standard deviations.

c. What is the value of[pic]? …………… Mark this value on the horizontal axis of your sketch. Shade the area to the left of this value.

d. Write the hypotheses for a significance test. H0: …………… and Ha: ……………

e. Using the 68-95-99.7 Rule and the placement of on the graph above, approximately how often would we can a sample proportion as low or lower than ours if the true proportion of residents 25 years old or

older really is 0.804? …………… So is getting 2,365 out of a sample of 3,000 usual or unusual? …………………

STA 2e: Sect. 9.2 Worksheet #2 Name ______________________________

P-values

1. A significance test looks for evidence against the ……………… hypothesis and in favor of the ……………… hypothesis.

2. The P-value is the probability that we would see a sample outcome as extreme or more extreme than the

actually observed outcome if the ………………… hypothesis really were true.

3. The smaller the P-value is, the ………………… is the evidence against H0 that is provided by the data.

(stronger/weaker)

4. Generally, we can think of P-values as follows:

If P-value < 0.10, then we have ………………… evidence against the null hypothesis.

If P-value < 0.05, then we have ………………… evidence against the null hypothesis.

If P-value < 0.01, then we have ………………… evidence against the null hypothesis.

5. Sometimes, before we do a significance test, we determine just how much evidence against H0 that we will

insist on. The decisive P-value is called the ……………………… …………… and is represented by the

Greek letter ……….

6. If H0: p = 0.5 and Ha: p > 0.5, then the significance test will be ……………-tailed. (right/left/two)

7. If H0: p = 0.5 and Ha: p ≠ 0.5, then the significance test will be ……………-tailed. (right/left/two)

8. If H0: p = 0.5 and Ha: p < 0.5, then the significance test will be ……………-tailed. (right/left/two)

9. For each of the following situations, determine if we have no evidence, some evidence, moderate evidence, or strong evidence against H0.

| |Hypotheses |Significance Level |P-value |Decision |

|a. |H0: p = 0.5 |α = 0.05 |P-value = 0.0325 | |

| |Ha: p > 0.5 | | | |

|b. |H0: p = 0.5 |α = 0.01 |P-value = 0.0325 | |

| |Ha: p > 0.5 | | | |

|c. |H0: p = 0.5 |α = 0.01 |P-value = 0.00325 | |

| |Ha: p > 0.5 | | | |

10. For each of the following, p = 0.5, n = 100. For all graphs, the P-value is the same. Which graph corresponds to each set of hypotheses?

a. H0: p = 0.5, Ha: p > 0.5 b. H0: p = 0.5, Ha: p < 0.5 c. H0: p = 0.5, Ha: p ≠ 0.5

To find the P-value, you must first find the standardized score. STA 2e: Sect. 9.2 Worksheet #2

The formula is:

For #11 – 16: If a die is fair, then the number 5 should occur 1/6 of the time. You have a die that you suspect is loaded so that the number 5 lands face up more often then expected. You roll the die 200 times and get 45 5's. Do we have evidence that the die is unfair (i.e. loaded)?

11. H0: ……………, Ha: ………………

12. mean = …………, standard deviation = ………… observation = [pic] = …………

–

13. Then standardized score = = ……………

14. The standardized score is a z-score. Mark this value on the graph at the right. Since we want to know if the number 5 appears more often than expected, we need to shade the area to the right of our z-score. Then find the area using Table A.

The area we find is the P-value. It’s value is ……………

15. Therefore, at α = 0.05 significance level, we would conclude which of the following:

a. There is strong evidence that the die is unfair.

b. There is moderate evidence that the die is unfair.

c. There is some evidence that the die is unfair.

d. There is insufficient evidence to determine whether the die is fair or unfair.

16. If we used α = 0.01significance level, which of the choices on #15 would be correct? ………

For #17 – 21: The White House press secretary comments that the president currently has a 72% favorable job approval rating. A pollster challenges this claim as being too high. His polling service has just conducted a random survey of 1000 people (calling both landline and cell phone numbers) and 680 people gave the president a favorable job approval rating. Do we have reason to doubt the press secretary?

17. H0: ……………, Ha: ………………

18. mean = …………, standard deviation = ………… observation = [pic] = …………

20. The standardized score is a z-score. Mark this value on the graph at the right. Since we want to know if the claim is too high, we think that the true proportion is actually less than 0.72. So we need to shade the area to the left of our z-score. Then find the area using Table A.

The area we find is the P-value. It’s value is ……………

21. Therefore, at α = 0.05 significance level, we would conclude which of the following:

a. There is strong evidence that the true favorable job approval rating is less than 72%.

b. There is moderate evidence that the true favorable job approval rating is less than 72%

c. There is some evidence that the true favorable job approval rating is less than 72%

d. There is insufficient evidence to determine whether the true favorable job approval rating is less than 72%.

STA 2e: Sect. 9.2 Worksheet #3 Name ______________________________

Tests of Proportions

For each of the following, (a) State null and alternative hypotheses.

(b) Describe the sampling distribution of the sample proportion.

(c) How extreme is the sample outcome? Find the P-value.

(d) Explain your conclusions in nontechnical language.

1. A 2009 nationwide random survey of 1500 adults asked the open-ended question, “What do you think is the most important problem facing this country today?” Sixty-nine percent responded with some form of economic problems (such as economy in general, unemployment/jobs, etc). Do these data provide good evidence that more than 2/3 of all adults believe that economic problems is the most important problem facing this country today?

a. H0: ………………

Ha: ………………

b. If the null hypothesis is true, so that p = ………, then follows a ……………… distribution with mean ………… and standard deviation ……………

P-value = …………

d. Conclusion: …………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

2. According to M&M/Mars Company, 20% of all Plain M&M’s produced are orange. You buy one large 56 oz bag of Plain M&M’s and find that it contains 355 orange M&M’s out of a total of 1,858 M&Ms. Do we have evidence that the company is lying about the proportion of orange M&M’s?

a. H0: ………………

Ha: ………………

b. If the null hypothesis is true, ………………………………………………………………………………………

…………………………………………………………………………………………………………………………

P-value = …………

d. Conclusion: …………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

STA 2e: Sect. 9.2 Worksheet #3 p.2

3. According to Sallie Mae, seventy-six percent of undergraduates had at least one credit card in 2004. In 2008, Sallie Mae took a survey of 280 undergraduates and found that 235 of these students had at least one credit card. Is there evidence that the proportion of all undergraduates with at least one credit card has increased in the four-year period?

a. [pic]………………

b. H0: ………………

Ha: ………………

c. …………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

d. standardized score =

P-value = …………

e. Conclusion: …………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

f. Note: A total of 1,200 students between the ages of 18 and 24 enrolled as undergraduate students in the spring of 2008 at public and private four-year colleges and universities was included in the credit bureau analysis. The survey was sent by mail and email in June and July to this group, resulting in an 8 percent response rate. The online survey was also sent to an additional 4,600 undergraduates enrolled in the spring at four-year colleges, ages 18 to 24, between August and September and received a 4 percent response rate. What type of bias does this survey suffer from?

4. The probability of getting a diamond when you draw one card from a deck of 52 playing cards is 0.25. Emma decides to see if this is true. She shuffles the cards, draws one card, and notes the suite. She then replaces the card and repeats the process for a total of 200 times. She draws a diamond 40 of these times. Is there evidence to suggest that the probability of getting a diamond when you draw one card is different from 0.25? Use a significance level of α = 0.05.

a. H0: ………………

Ha: ………………

b. …………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

c. standardized score =

P-value = …………

d. Conclusion: …………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………

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