9. Hypothesis Testing for One Sample

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?

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

A city claims that $30\%$ of households recycle regularly. A researcher surveys $200$ households to see if the true proportion is different and finds that $78$ recycle regularly. Use $\alpha =0.05$ to test the claim.

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?

Perception and Reality In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won (based on data from ICR Survey Research Group). Use a 0.05 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

use the figure at the left, which suggests what adults think about protecting the environment.

[Image]

Are People Concerned About Protecting the Environment? You interview a random sample of 100 adults. The results of the survey show that 58% of the adults said they live in ways that help protect the environment some of the time. At α=0.05, can you reject the claim that at least 64% of adults make an effort to live in ways that help protect the environment some of the time?

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:

$\alpha =0.10$, $n=100$, $x=42$, & claim is $p=0.25$

"[NW] Small-Sample Hypothesis TestProfessors 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 α = 0.05 level of significance?

a. State the appropriate null and alternative hypotheses.

Source: Kirk, Honey and Lerma, Diane, “Reading Your Way to Success in Mathematics: A Paired Course of Developmental Mathematics and Reading.” MathAMATYC Educator, Vol. 1 No. 2, 2010."

"Living Alone? In 2000, 58% of females aged 15 or 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 and older and finds that 285 are living alone. Is there sufficient evidence to conclude the proportion has changed since 2000?

a. What is the variable of interest in this study? Is it qualitative with two outcomes? Explain."

Explain how to test a population proportion p.

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?

"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.

a. Suppose the likelihood of predicting the direction of the stock market (increasing or decreasing) in any given year is 0.50. Decide on the appropriate null and alternative hypotheses to test whether the outcome of the Super Bowl can be used to predict the direction of the stock market.

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In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p ≥0.48, α=0.08. Sample statistics: p_hat = 0.40, n=90

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?

Using Technology

In Exercises 5–8, identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section. Use a 0.05 significance level and answer the following:

a. Is the test two-tailed, left-tailed, or right-tailed?

b. What is the test statistic?

c. What is the P-value?

d. What is the null hypothesis, and what do you conclude about it?

e. What is the final conclusion?

Adverse Reactions to Drug The drug Lipitor (atorvastatin) is used to treat high cholesterol. In a clinical trial of Lipitor, 47 of 863 treated subjects experienced headaches (based on data from Pfizer). The accompanying TI-83/84 Plus calculator display shows results from a test of the claim that fewer than 10% of treated subjects experience headaches.