9. Hypothesis Testing for One Sample

Self-InjuryAccording to the article “Self-injurious Behaviors in a College Population,” 17% of undergraduate or graduate students have had at least one incidence of self-injurious behavior. The researchers conducted a survey of 40 college students who reported a history of emotional abuse and found that 12 of them have had at least one incidence of self-injurious behavior. What do the results of this survey tell you about college students who report a history of emotional abuse?

Taught Enough Math? In 1994, 52% of parents with 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 with 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 α = 0.05 level of significance?

Sneeze According to work done by Nick Wilson of Otago University Wellington, the proportion of individuals who cover their mouth when sneezing is 0.733. As part of a school project, Mary decides to confirm this by observing 100 randomly selected individuals sneeze and finds that 78 covered their mouth when sneezing.

a. What are the null and alternative hypotheses for Mary’s project?

b. Verify the requirements that allow use of the normal model to test the hypotheses are satisfied.

c. Does the sample evidence contradict Professor Wilson’s findings?

Emergency Room The proportion of patients who visit the emergency room (ER) and die within the year is 0.05. Source: SuperFreakonomics. Suppose a hospital administrator is concerned that his ER has a higher proportion of patients who die within the year. In a random sample of 250 patients who have visited the ER in the past year, 17 have died. Should the administrator be concerned?

Teen Prayer In 1995, 40% of adolescents stated they prayed daily. A researcher wants to know whether this percentage has become higher since then. He surveys 40 adolescents and finds that 18 pray on a daily basis. Is there enough evidence to support the proportion of adolescents who pray daily has increased at the α = 0.05 level of significance?

Course RedesignPass rates for Intermediate Algebra at a community college are 52.6%. In an effort to improve pass rates in the course, faculty of a community college develop a mastery-based learning model where course content is delivered in a lab through a computer program. The instructor serves as a learning mentor for the students. Of the 480 students who enroll in the mastery-based course, 267 pass.

b. At the 0.01 level of significance, decide whether the sample evidence suggests the mastery-based learning model improved pass rates.

In Problems 7–12, 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 = 0.3versusH1: p > 0.3n = 200;x = 75;α = 0.05

In Problems 7–12, 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 = 0.4versusH1: p ≠ 0.4n = 1000;x = 420;α = 0.01

You Explain It! ESPSuppose 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 ≈ 0.083, the rate one would expect from simply guessing.

b. Suppose the individual was able to guess nine 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.

Small Sample Hypothesis Test: Super Bowl InvestingFrom Super Bowl I (1967) through Super Bowl XXXI (1997), the stock market increased if an NFL team won the Super Bowl and decreased if an AFL team won. This condition held 28 out of 31 years.

b. Use the binomial probability distribution to determine the P-value for the hypothesis test from part (a).

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?

In Problems 1–6, test the hypothesis using the P-value approach. Be sure to verify the requirements of the test.

H₀: p = 0.3 versus H₁: p > 0.3

n = 200; x = 75; α = 0.05

You Explain It! Stock Analyst 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, 48 of the companies were considered winners; that is, they outperformed other companies. 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.

Blind Emotion [See Problem 11 in Section 10.2A.] When the area of the brain responsible for vision is destroyed, individuals experience cortical blindness. Patients with cortical blindness are unaware of any visual stimulus including light. In a 52-year-old male patient with cortical blindness (as a result of two strokes within a 38-day timeframe), a series of visual stimuli were presented on a computer screen. The patient was given two choices for each stimulus and asked to report what was on the screen. The patient’s responses were recorded by an individual who could not see the contents on the screen.

c. The researchers wanted to determine if the patient could identify other facial characteristics. They randomly showed male or female faces and asked the patient to identify the gender. The patient was correct in 89 of 200 trials. What does this suggest?

Statistics in the Media A headline read, “More Than Half of Americans Say Federal Taxes Too High.” The headline was based on a random sample of 1026 adult Americans in which 534 stated the amount of federal tax they have to pay is too high. Is this an accurate headline?