Sullivan Statistics Survey: Political Philosophy According to Gallup, 21% of Americans

When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is and we reject the null hypothesis.

True or False: When testing a hypothesis using the Classical Approach, if the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis.

True or False: When testing a hypothesis using the P-value Approach, if the P-value is large, we reject the null hypothesis

Determine the critical value for a right-tailed test regarding a population proportion at the a = 0.01 level of significance.

Determine the critical value for a left-tailed test regarding a population proportion at the a = 0.1 level of significance.

Determine the critical value for a two-tailed test regarding a population proportion at the a = 0.05 level of significance.

In Problems 712, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test H0: p = 0.3 versus H1: p 7 0.3 n = 200; x = 75; a = 0.05

In Problems 712, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test H0: p = 0.6 versus H1: p 6 0.6 n = 250; x = 124; a = 0.01

In Problems 712, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test H0: p = 0.55 versus H1: p 6 0.55 n = 150; x = 78; a = 0.1

In Problems 712, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test H0: p = 0.25 versus H1: p 6 0.25 n = 400; x = 96; a = 0.1

In Problems 712, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the testH0: p = 0.9 versus H1: p 0.9 n = 500; x = 440; a = 0.05

In Problems 712, test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test H0: p = 0.4 versus H1: p 0.4 n = 1000; x = 420; a = 0.01

You Explain It! Stock Analyst Some have argued that throwing darts at the stock pages to decide which companies to invest in could be a successful stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 100 companies to invest in. After 1 year, 53 of the companies were considered winners; that is, they outperformed other companies in the same investment class. To assess whether the dart-picking strategy resulted in a majority of winners, the researcher tested H0: p = 0.5 versus H1: p > 0.5 and obtained a P-value of 0.2743. Explain what this P-value means and write a conclusion for the researcher.

You Explain It! ESP Suppose an acquaintance claims to have the ability to determine the birth month of randomly selected individuals. To test such a claim, you randomly select 80 individuals and ask the acquaintance to state the birth month of the individual. If the individual has the ability to determine birth month, then the proportion of correct birth months should exceed 1 12 , the rate one would expect from simply guessing. (a) State the null and alternative hypotheses for this experiment. (b) Suppose the individual was able to guess 9 correct birth months. The P-value for such results is 0.1726. Explain what this P-value means and write a conclusion for the test.

Lipitor The drug Lipitor is meant to reduce total cholesterol and LDL cholesterol. In clinical trials, 19 out of 863 patients taking 10 mg of Lipitor daily complained of flulike symptoms. Suppose that it is known that 1.9% of patients taking competing drugs complain of flulike symptoms. Is there evidence to conclude that more than 1.9% of Lipitor users experience flulike symptoms as a side effect at the a = 0.01 level of significance?

Nexium Nexium is a drug that can be used to reduce the acid produced by the body and heal damage to the esophagus due to acid reflux. The manufacturer of Nexium claims that more than 94% of patients taking Nexium are healed within 8 weeks. In clinical trials, 213 of 224 patients suffering from acid reflux disease were healed after 8 weeks. Test the manufacturers claim at the a = 0.01 level of significance.

Worried about Retirement? In April 2009, the Gallup organization surveyed 676 adults aged 18 and older and found that 352 believed they would not have enough money to live comfortably in retirement. Does the sample evidence suggest that a majority of adults in the United States believe they will not have enough money in retirement? Use the a = 0.05 level of significance.

Ready for College? There are two major college entrance exams that a majority of colleges accept for admission, the SAT and ACT. ACT looked at historical records and established 21 as the minimum score on the ACT reading portion of the exam for a student to be considered prepared for social science in college. Note: Being prepared means there is a 75% probability of successfully completing a social science in college. An official with the Illinois State Department of Education wonders whether a majority of the students in her state are prepared to take social science. She obtains a simple random sample of 500 records of students who have taken the ACT and finds that 269 are prepared for social science in college (that is, scored at least 21 on the reading portion of the ACT). Does this represent significant evidence that a majority of the students in the state of Illinois are prepared for social science in college upon graduation? Use the a = 0.05 level of significance. Source: ACT High School Profile Report.

Critical Job Skills In August 2003, 56% of employed adults in the United States reported that basic mathematical skills were critical or very important to their job. The supervisor of the job placement office at a 4-year college thinks this percentage has increased due to increased use of technology in the workplace. He takes a random sample of 480 employed adults and finds that 297 of them feel that basic mathematical skills are critical or very important to their job. Is there sufficient evidence to conclude that the percentage of employed adults who feel basic mathematical skills are critical or very important to their job has increased at the a = 0.05 level of significance?

Critical Job Skills In August 2003, 56% of employed adults in the United States reported that basic mathematical skills were critical or very important to their job. The supervisor of the job placement office at a 4-year college thinks this percentage has increased due to increased use of technology in the workplace. He takes a random sample of 480 employed adults and finds that 297 of them feel that basic mathematical skills are critical or very important to their job. Is there sufficient evidence to conclude that the percentage of employed adults who feel basic mathematical skills are critical or very important to their job has increased at the a = 0.05 level of significance?

Taught Enough Math? In 1994, 52% of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in 1994? Use the a = 0.05 level of significance.? Source: Based on Reality Check: Are Parents and Students Ready for More Math and Science? Public Agenda, 2006.

Living Alone? In 2000, 58% of females aged 15 years of age and older lived alone, according to the U.S. Census Bureau. A sociologist tests whether this percentage is different today by conducting a random sample of 500 females aged 15 years of age and older and finds that 285 are living alone. Is there sufficient evidence at the a = 0.1 level of significance to conclude the proportion has changed since 2000?

Quality of Education In August 2002, 47% of parents who had children in grades K12 were satisfied with the quality of education the students receive. In September 2010, the Gallup organization conducted a poll of 1013 parents who have children in grades K12 and asked if they were satisfied with the quality of education the students receive. Of the 1013 surveyed, 437 indicated that they were satisfied. Construct a 95% confidence interval to assess whether this represents evidence that parents attitudes toward the quality of education in the United States has changed since August 2002.

Infidelity According to menstuff.org, 22% of married men have strayed at least once during their married lives. (a) Describe how you might go about administering a survey to assess the accuracy of this statement. (b) A survey of 500 married men results in 122 indicating that they have strayed at least once during their married life. Construct a 95% confidence interval for the population proportion of married men who have strayed. Use this interval to assess the accuracy of the statement made by menstuff.org.

Talk to the Animals In a survey conducted by the American Animal Hospital Association, 37% of respondents stated that they talk to their pets on the answering machine or telephone. A veterinarian found this result hard to believe, so he randomly selected 150 pet owners and discovered that 54 of them spoke to their pet on the answering machine or telephone. Does the veterinarian have the right to be skeptical? Use a 0.05 level of significance.

Eating Salad According to a survey conducted by the Association for Dressings and Sauces (this is an actual association!), 85% of American adults eat salad at least once a week. A nutritionist suspects that the percentage is higher than this. She conducts a survey of 200 American adults and finds that 171 of them eat salad at least once a week. Conduct the appropriate test that addresses the nutritionists suspicions. Use a 0.1 level of significance.

Small-Sample Hypothesis Test Professors Honey Kirk and Diane Lerma of Palo Alto College developed a learning community curriculum that blended the developmental mathematics and the reading curriculum with a structured emphasis on study skills. In a typical developmental mathematics course at Palo Alto College, 50% of the students complete the course with a letter grade of A, B, or C. In the experimental course, of the 16 students enrolled, 11 completed the course with a letter grade of A, B, or C. Do you believe the experimental course was effective at the a = 0.05 level of significance? Source: Kirk, Honey & Lerma, Diane, Reading Your Way to Success in Mathematics: A Paired Course of Developmental Mathematics and Reading. MathAMATYC Educator, Vol.1. No. 2, 2010.

Small-Sample Hypothesis Test In 1997, 4% of mothers smoked more than 21 cigarettes during their pregnancy. An obstetrician believes that the percentage of mothers who smoke 21 cigarettes or more is less than 4% today. She randomly selects 120 pregnant mothers and finds that 3 of them smoked 21 or more cigarettes during pregnancy. Does the sample data support the obstetricians belief? Use the a = 0.05 level of significance.

Small-Sample Hypothesis Test According to the U.S. Census Bureau, in 2009, 2.1% of Americans worked at home. An economist believes that the percentage of Americans working at home has increased since then. He randomly selects 150 working Americans and finds that 6 of them work at home. Do a higher proportion of Americans work at home? Use the a = 0.05 level of significance.

Small-Sample Hypothesis Test According to the U.S. Census Bureau, in 2009, 7.8% of Americans had a travel time to work of more than 60 minutes. An urban economist believes that the percentage of Americans who have a travel time to work of more than 60 minutes has increased since then. She randomly selects 80 working Americans and finds that 11 of them have a travel time to work that is more than 60 minutes. Test the urban economists belief at the a = 0.1 level of significance.

Statistics in the Media One of the more popular statistics reported in the media is the presidents job approval rating. The approval rating is reported as the proportion of Americans who approve of the job that the sitting president is doing and is typically based on a random sample of registered voters. (a) This proportion tends to fluctuate from week to week. Name some reasons for the fluctuation in the statistic. (b) A recent article had the headline Obama Ratings Rise. This headline was written because a December 2010 poll showed President Obamas approval rating to be 0.48 (48%). A poll the following January based on 1500 randomly selected Americans showed that 747 approved of the job Obama was doing. Do the results of the January poll indicate that the proportion of Americans who approve of the job Obama is doing is significantly higher than Decembers level? Explain

Statistics in the Media In November 2009, 65% (0.65) of Americans favored the death penalty for a person convicted of murder. In November 2010, 1025 adult Americans were asked by the Gallup organization, Are you in favor of the death penalty for a person convicted of murder? Of the 1025 adults surveyed, 653 responded yes. The headline in the article reporting the survey results stated, Fewer Americans Favor the Death Penalty. Use a test of significance to support or refute this headline.

Sullivan Statistics Survey: Political Philosophy According to Gallup, 21% of Americans consider themselves to be liberal. The Sullivan Statistics Survey asked respondents to disclose their political philosophy: Conservative, Liberal, Moderate. Treat the results of the Sullivan Statistics Survey as a random sample of adult Americans. Do the survey results suggest the proportion is higher than that reported by Gallup? Use an a = 0.1 level of significance.

Sullivan Statistics Survey: Smokers According to the Centers for Disease Control, in 2008, 21% of adult Americans age 18 or older were current smokers. Treat the results of the Sullivan Statistics Survey as a random sample of adult Americans. Do the survey results suggest that the proportion of adult Americans who are current smokers has declined? Use an a = 0.05 level of significance

Are Spreads Accurate? For every NFL game there is a team that is expected to win by a certain number of points. In betting parlance, this is called the spread. For example, if the Chicago Bears are expected to beat the Green Bay Packers by 3 points, a sports bettor would say, Chicago is minus three. So, if the Bears lose to Green Bay, or do not win by more than 3 points, a bet on Chicago would be a loser. If point spreads are accurate, we would expect about half of all games played to result in the favored team winning (beating the spread) and about half of all games to result in the team favored to not beat the spread. The following data represent the results of 45 randomly selected games where a 0 indicates the favored team did not beat the spread and a 1 indicates the favored team beat the spread. Do the data suggest that sport books establish accurate spreads? 00 000 1 0 0 1 0 00 010 11 001 00 111 00 110 10 010 01 101 00 111 Source: http://www.vegasinsider.com

Accept versus Do Not Reject In the United States, historically, 40% of registered voters are Republican. Suppose you obtain a simple random sample of 320 registered voters and find 142 registered Republicans. (a) Consider the hypotheses H0: p = 0.4 versus H1: p 7 0.4. Explain what the researcher would be testing. Perform the test at the a = 0.05 level of significance. Write a conclusion for the test. (b) Consider the hypotheses H0: p = 0.41 versus H1: p 7 0.41. Explain what the researcher would be testing. Perform the test at the a = 0.05 level of significance. Write a conclusion for the test. (c) Consider the hypotheses H0: p = 0.42 versus H1: p 7 0.42. Explain what the researcher would be testing. Perform the test at the a = 0.05 level of significance. Write a conclusion for the test. (d) Based on the results of parts (a)(c), write a few sentences that explain the difference between accepting the statement in the null hypothesis versus not rejecting the statement in the null hypothesis.

Interesting Results Suppose you wish to find out the answer to the age-old question, Do Americans prefer Coke or Pepsi? You conduct a blind taste test in which individuals are randomly asked to drink one of the colas first, followed by the other cola, and then asked to disclose which drink they prefer. Results of your taste test indicate that 53 of 100 individuals prefer Pepsi. (a) Conduct a hypothesis test (preferably using technology) H0: p = p0 versus H1: p p0 for p0 = 0.42, 0.43, 0.44, p , 0.64 at the a = 0.05 level of significance. For which values of p0 do you not reject the null hypothesis? What do each of the values of p0 represent? (b) Construct a 95% confidence interval for the proportion of individuals who prefer Pepsi. (c) Suppose you changed the level of significance in conducting the hypothesis test to a = 0.01? What would happen to the range of values of p0 for which the null hypothesis is not rejected? Why does this make sense?

Simulation Simulate drawing 100 simple random samples of size n = 40 from a population whose proportion is 0.3. (a) Test the null hypothesis H0: p = 0.3 versus H1: p 0.3 for each simulated sample. (b) If we test the hypothesis at the a = 0.1 level of significance, how many of the 100 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? (d) How do we know that a rejection of the null hypothesis results in making a Type I error in this situation?

Simulation Simulate drawing 100 simple random samples of size n = 15 from a population whose proportion is 0.2. (a) Test the null hypothesis H0: p = 0.2 versus H1: p 0.2 for each simulated sample. (b) If we test the hypothesis at the a = 0.1 level of significance, how many of the 100 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? What might account for any discrepancies? (d) How do we know that a rejection of the null hypothesis results in making a Type I error in this situation? 4

Putting It Together: Lupus Based on historical birthing records, the proportion of males born worldwide is 0.51. In other words, the commonly held belief that boys are just as likely as girls is false. Systematic lupus erythematosus (SLE), or lupus for short, is a disease in which ones immune system attacks healthy cells and tissue by mistake. It is well known that lupus tends to exist more in females than in males. Researchers wondered, however, if families with a child who had lupus had a lower ratio of males to females than the general population. If this were true, it would suggest that something happens during conception that causes males to be conceived at a lower rate when the SLE gene is present. To determine if this hypothesis is true, the researchers obtained records of families with a child who had SLE. A total of 23 males and 79 females were found to have SLE. The 23 males with SLE had a total of 23 male siblings and 22 female siblings. The 79 females with SLE had a total of 69 male siblings and 80 female siblings. Source: L.N. Moorthy, M.G.E. Peterson, K.B. Onel, and T.J.A. Lehman. Do Children with Lupus Have Fewer Male Siblings Lupus 2008 17:128131, 2008. (a) Explain why this is an observational study. (b) Is the study retrospective or prospective? Why? (c) There are a total of 23 + 69 = 92 male siblings in the study. How many female siblings are in the study? (d) Draw a relative frequency bar graph of gender of the siblings. (e) Find a point estimate for the proportion of male siblings in families where one of the children has SLE. (f) Does the sample evidence suggest that the proportion of male siblings in families where one of the children has SLE is less than 0.51, the accepted proportion of males born in the general population? Use the a = 0.05 level of significance. (g) Construct a 95% confidence interval for the proportion of male siblings in a family where one of the children has SLE.

character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill or a hinderer toy prevent the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. The baby was then placed in front of each toy and allowed to choose which toy he or she wished to play with. In 14 of the 16 cases, the baby chose the helper toy. Source: J. Kiley Hamlin et al., Social Evaluation by Preverbal Infants. Nature, Nov. 2007. (a) Why is it important to randomly expose the baby to the helper or hinderer toy first? (b) What would be the appropriate null and alternative hypotheses if the researcher is attempting to show that babies prefer helpers over hinderers? (c) Use the binomial probability formula to determine the P-value for this test. (d) In testing 12 six-month-old babies, all 12 preferred the helper toy. The P-value was reported as 0.0002. Interpret this result.

Explain what a P-value is. What is the criterion for rejecting the null hypothesis using the P-value approach?

Suppose we are testing the hypothesis H0: p = 0.3 versus H1: p > 0.3 and we find the P-value to be 0.23. Explain what this means. Would you reject the null hypothesis? Why?

Suppose we are testing the hypothesis H0: p = 0.65 versus H1: p 0.65 and we find the P-value to be 0.02. Explain what this means. Would you reject the null hypothesis? Why?

Discuss the advantages and disadvantages of using the Classical Approach to hypothesis testing. Discuss the advantages and disadvantages of using the P-value approach to hypothesis testing.

The headline reporting the results of a poll conducted by the Gallup organization stated Majority of Americans at Personal Best in the Morning. The results indicated that a survey of 1100 Americans resulted in 55% stating they were at their personal best in the morning. The polls results were reported with a margin of error of 3%. Explain why the Gallup organizations headline is accurate.

Explain what statistical significance means.