9. Hypothesis Testing for One Sample

"[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."

"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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"Put the following P-values in order from weakest to strongest in terms of evidence against the statement in the null hypothesis.

a. 0.139

b. 0.083

c. 0.091

d. 0.005

e. 0.019"

"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."

"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?

b. What is the goal of the research?"

"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?

c. What are the null and alternative hypotheses?"

"Cramer Correct Less Than Half the Time? [See Problem 3 in Section 10.2A.] The website pundittracker.com keeps track of predictions made by individuals in finance, politics, sports, and entertainment. Jim Cramer is a famous TV financial personality and author. Pundittracker monitored 678 of his stock predictions (such as a recommendation to buy the stock) and found that 320 were correct predictions. Treat these 678 predictions as a random sample of all of Cramer’s predictions.

c. Verify the normal model may be used to determine the P-value for this hypothesis test.

"Political Pundits [See Problem 4 in Section 10.2A.] In his book, “The Signal and the Noise,” Nate Silver analyzed 733 predictions made by experts regarding political events. Of the 733 predictions, 338 were mostly true.

e. Determine the P-value based on the model from part (d).

f. Interpret the P-value."

"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."

"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.

a. An initial baseline test was conducted by presenting black squares and circles on the white background of the screen. The patient correctly “guessed” the item on the screen in 90 of the 200 trials. Researchers concluded the patient’s performance was not statistically different from chance. Explain what this means using the language of hypothesis testing.

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"Simulation Simulate drawing 100 simple random samples of size n = 40 from a population whose proportion is 0.3.

d. How do we know that a rejection of the null hypothesis results in making a Type I error in this situation?"

True or False: Sample evidence can prove a null hypothesis is true.