Math 121 Chapter 8
You roll a six-sided die 56 times and land on an ace (a one) 8 times. You want to test the hypothesis that the die does not come up with an ace one-sixth of the time. Determine the null hypothesis.
H0: p = 1/6
If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?
No, if the p-value is sufficiently small, the null hypothesis is unlikely to be true, but unlikely is not the same as impossible.
Is it acceptable practice to look at your research results, note the direction of the difference, and then make the alternative hypothesis one-sided in order to achieve a significant difference? Explain.
No. Changing the alternative hypothesis is considered cheating.
A person spinning a 1962 penny gets 10 heads out of 50 spins. Because she gets a p-value of 0.00002, she says she has proved the coin is biased. What is the flaw in the statement and how should it be corrected?
The flaw is that the spinner said she proved the coin is biased. Hypothesis tests are not used to prove ideas.
A weight-loss diet claims that it causes weight loss by eliminating carbohydrates (breads and starches) from the diet. To test this claim, researchers randomly assign overweight subjects to two groups. Both groups eat the same amount of calories, but one group eats almost no carbs, and the other group includes carbs in their meals. After 2 months, the researchers test the claim that the no-carb diet is better than the usual diet. They record the proportion of each group that lost more than 5% of their initial weight. They then announce that they failed to reject the null hypothesis. Which of the following are valid interpretations of the researchers' findings?
There were no significant differences in effectiveness between the no-carb diet and the carb diet. The researchers did not see enough evidence to conclude that the no-carb diet was more effective.
About 32% of the population in a large region are between the ages of 40 and 65, according to the country's census. However, only 3% of the 2200 employees at a company in the region are between the ages of 40 and 65. Lawyers might argue that if the company hired people regardless of their age, the distribution of ages would be the same as though they had hired people at random from the surrounding population. Check whether the conditions for using the one-proportion z-test are met.
Yes, all conditions are satisfied or can be reasonably assumed.
When, in a criminal court, a defendant is found "not guilty," is the court saying with certainty that he or she is innocent? Explain.
No, the court is only saying that there is not enough evidence to show guilt.
In statistical inference, measurements are made on a _________________and generalizations are made to a _____________.
In statistical inference, measurements are made on a sample and generalizations are made to a population.
When a person stands trial for murder, the jury is instructed to assume that the defendant is innocent. Is this claim of innocence an example of a null hypothesis, or is it an example of an alternative hypothesis?
It is a null hypothesis, since it is assumed to be true until evidence can prove otherwise.
A new drug is being tested to see whether it can reduce the chance of an asthma attack in people who have had an asthma attack in the last week. The rate of asthma attack in the population of concern is 0.13. The null hypothesis is that p (the population proportion using the new drug that have an asthma attack) is 0.13. What is the correct alternative hypothesis?
P < or less than 0.13
The null hypothesis is always a statement about what?
The null hypothesis is always a statement about a population parameter.
Suppose a researcher is testing someone to see if he or she can tell Soda X from Soda Y, and the researcher is using 20 trials, half with Soda X and half with Soda Y. The null hypothesis is that the person is guessing. The alternative is one-sided, Upper H Subscript a: p 0greater than0.5. The person gets 12 right out of 20. The p-value comes out to be 0.187. Explain the meaning of the p-value.
The probability that a person will get 12 or more right, if the person is truly guessing, is about 19%.
The null hypothesis on true/false tests is that the student is guessing, and the proportion of right answers is 0.50. A student taking a five-question true/false quiz gets 4 right out of 5. She says that this shows that she knows the material, because the one-tailed p-value from the one-proportion z-test is 0.090, and she is using a significance level of 0.10. What is wrong with her approach?
The sample size is not large enough to use the one-proportion z-test. To use that test, the tester must expect at least 10 successes and at least 10 failures.
A magazine advertisement claims that wearing a magnetized bracelet will reduce arthritis pain in those who suffer from arthritis. A medical researcher tests this claim with 233 arthritis sufferers randomly assigned to wear either a magnetized bracelet or a placebo bracelet. The researcher records the proportion of each group who report relief from arthritis pain after 6 weeks. After analyzing the data, he fails to reject the null hypothesis. What are valid interpretations of his findings?
There were no statistically significant differences between the magnetized bracelets and the placebos in reducing arthritis pain. There's insufficient evidence that the magnetized bracelets are effective at reducing arthritis pain.
A psychologist is interested in testing whether offering students a financial incentive improves their video-game-playing skills. She collects data and performs a hypothesis test to test whether the probability of getting to the highest level of a video game is greater with a financial incentive than without. Her null hypothesis is that the probability of getting to this level is the same with or without a financial incentive. The alternative is that this probability is greater. She gets a p-value from her hypothesis test of 0.003. What is the best interpretation of the p-value?
The p-value is the probability of getting a result as extreme as or more extreme than the one obtained, assuming that financial incentives are not effective in this context.
When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value if the sample proportions are close together or if they are far apart? Explain.
The p-value will be smaller if the sample proportions are far apart because a larger difference results in a larger absolute value of the numerator of the test statistic.
When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.
The p-value will be smaller with a larger sample size because a larger sample results in a smaller standard error, which is the denominator of the test statistic.
Assume that the proportion of people who live after suffering an aneurysm is 0.86. Suppose there is a new medicine that is used to increase the survival rate. Use the parameter p to represent the population proportion of people who survive after an aneurysm. For a hypothesis test of the medicine's effectiveness, researchers use a null hypothesis of pequals0.86. What is the correct alternative hypothesis?
p > or greater than 0.86
If we do not reject the null hypothesis, is it valid to say that we accept the null hypothesis? Why or why not?
No, we have only shown that we do not have enough evidence to reject it.
A 2003 study of dreaming found that out of a random sample of 102 people, 80 reported dreaming in color. However, the rate of reported dreaming in color that was established in the 1940s was 0.28. Check to see whether the conditions for using a one-proportion z-test are met assuming the researcher wanted to test to see if the proportion dreaming in color had changed since the 1940s.
Yes, all the conditions are met.