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Chapter 4: Problem 143

Clicker Questions A statistics instructor would like to ask "clicker" questions that about \(80 \%\) of her students in a large lecture class will get correct. A higher proportion would be too easy and a lower proportion might discourage students. Suppose that she tries a sample of questions and receives 76 correct answers and 24 incorrect answers among 100 responses. The hypotheses of interest are \(H_{0}: p=0.80\) vs \(H_{a}: p \neq 0.80 .\) Discuss whether or not the methods described below would be appropriate ways to generate randomization samples in this setting. Explain your reasoning in each case. (a) Sample 100 answers (with replacement) from the original student responses. Count the number of correct responses. (b) Sample 100 answers (with replacement) from a set consisting of 8 correct responses and 2 incorrect responses. Count the number of correct responses.

Short Answer

Expert verified
Method (a) is not an appropriate means of generating randomization samples due to potential leaning towards observed data. Method (b), however, is deemed suitable since it mirrors the null hypothesis, leading to a balanced and fair comparison for hypothesis testing.

Step by step solution

01

Evaluation of Method (a)

Method (a) suggests sampling 100 answers with replacement from the original student responses and counting the number of correct responses. This method would follow the original student response proportions (76% correct, 24% incorrect), which may not be an appropriate reflection of the null hypothesis of 80% correctness, since it might lean towards the observed data. This method may thus not be an appropriate way to generate randomization samples.
02

Evaluation of Method (b)

Method (b) proposes sampling 100 answers with replacement from a set with 8 correct responses and 2 incorrect responses and then counting the correct responses. This method mirrors the null hypothesis that 80% of responses are correct and 20% are incorrect, providing a better reflection of the expected distributions under the null hypothesis than Method (a). This method would provide a balanced and fair comparison for hypothesis testing and can therefore be considered an appropriate way to generate randomization samples.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistics Education
Statistics education plays a critical role in helping students understand how to interpret data and make informed decisions based on evidence. It is not just about learning formulas or performing calculations; it involves a deeper understanding of the principles that guide data analysis. In the context of the exercise, it demonstrates how hypothesis testing can be applied in an educational setting.

Teaching students about hypothesis testing involves guiding them to formulating a hypothesis, an essential skill in statistics.
  • They need to understand the null hypothesis ( $H_0$ ), which often assumes no effect or no difference.
  • Equally important is the alternative hypothesis ( $H_a$ ), which is typically what researchers are trying to prove.
Hypothesis testing helps students learn how to systematically compare observed results with expected outcomes, fostering critical thinking. This can then be applied across various fields of study, illustrating the versatility and power of statistical analysis.
Randomization Techniques
Randomization techniques are crucial when conducting statistical experiments or sampling. These techniques ensure that every sample has an equal chance of being selected, which helps eliminate bias and increase the validity of the results.

In the given exercise, the instructor employs randomization to test if the proportion of correct answers aligns with the expected 80%.
  • Method (a) involves sampling from real student responses, repeating data collection.
  • Method (b) involves sampling from a synthetic dataset that reflects the hypothesized distribution.
Random sampling strives to ensure that every sample is a true representation of the population. Method (b) effectively mirrors the null hypothesis by using a fair mix of correct and incorrect responses, which facilitates a better assessment of the hypothesis being tested.
Null Hypothesis
The null hypothesis is a foundational concept in statistics, central to hypothesis testing. It represents a statement, usually of no effect or no change, that we seek to reject or fail to reject based on the data.

In the exercise, the null hypothesis is that 80% of the instructor鈥檚 students would answer correctly ( $H_0: p = 0.80$ ). It is essential to determine if the observed 76% reflects this hypothesis or if it suggests a significant difference.
  • The null hypothesis provides a baseline for comparison.
  • By comparing the null with the observed data, statisticians can infer whether data supports or refutes the null.
Evaluating the null hypothesis involves using statistical tests to see if there is enough evidence to suggest that observed data significantly deviates from the expected distribution. This involves calculating sample means, standard deviations, and other statistical measures. If the data shows a significant difference, the null hypothesis is rejected in favor of the alternative hypothesis.

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Most popular questions from this chapter

Arsenic in Chicken Data 4.5 on page 228 discusses a test to determine if the mean level of arsenic in chicken meat is above 80 ppb. If a restaurant chain finds significant evidence that the mean arsenic level is above \(80,\) the chain will stop using that supplier of chicken meat. The hypotheses are $$ \begin{array}{ll} H_{0}: & \mu=80 \\ H_{a}: & \mu>80 \end{array} $$ where \(\mu\) represents the mean arsenic level in all chicken meat from that supplier. Samples from two different suppliers are analyzed, and the resulting p-values are given: Sample from Supplier A: p-value is 0.0003 Sample from Supplier B: p-value is 0.3500 (a) Interpret each p-value in terms of the probability of the results happening by random chance. (b) Which p-value shows stronger evidence for the alternative hypothesis? What does this mean in terms of arsenic and chickens? (c) Which supplier, \(\mathrm{A}\) or \(\mathrm{B}\), should the chain get chickens from in order to avoid too high a level of arsenic?

Flipping Coins We flip a coin 150 times and get 90 heads, so the sample proportion of heads is \(\hat{p}=90 / 150=0.6 .\) To test whether this provides evidence that the coin is biased, we create a randomization distribution. Where will the distribution be centered? Why?

Red Wine and Weight Loss Resveratrol, an ingredient in red wine and grapes, has been shown to promote weight loss in rodents. A recent study \(^{19}\) investigates whether the same phenomenon holds true in primates. The grey mouse lemur, a primate, demonstrates seasonal spontaneous obesity in preparation for winter, doubling its body mass. A sample of six lemurs had their resting metabolic rate, body mass gain, food intake, and locomotor activity measured for one week prior to resveratrol supplementation (to serve as a baseline) and then the four indicators were measured again after treatment with a resveratrol supplement for four weeks. Some p-values for tests comparing the mean differences in these variables (before vs after treatment) are given below. In parts (a) to (d), state the conclusion of the test using a \(5 \%\) significance level, and interpret the conclusion in context. (a) In a test to see if mean resting metabolic rate is higher after treatment, \(p=0.013\). (b) In a test to see if mean body mass gain is lower after treatment, \(p=0.007\) (c) In a test to see if mean food intake is affected by the treatment, \(p=0.035\). (d) In a test to see if mean locomotor activity is affected by the treatment, \(p=0.980\) (e) In which test is the strongest evidence found? The weakest? (f) How do your answers to parts (a) to (d) change if the researchers make their conclusions using a stricter \(1 \%\) significance level? (g) For each p-value, give an informal conclusion in the context of the problem describing the level of evidence for the result. (h) The sample only included six lemurs. Do you think that we can generalize to the population of all lemurs that body mass gain is lower on average after four weeks of a resveratrol supplement? Why or why not?

Beer and Mosquitoes Does consuming beer attract mosquitoes? A study done in Burkino Faso, Africa, about the spread of malaria investigated the connection between beer consumption and mosquito attraction. \(^{9}\) In the experiment, 25 volunteers consumed a liter of beer while 18 volunteers consumed a liter of water. The volunteers \({ }^{8}\) Bouchard, M., Bellinger, D., Wright, \(\mathrm{R}\), and Weisskopf, M. "Attention-Deficit/Hyperactivity Disorder and Urinary Metabolites of Organophosphate Pesticides," Pediatrics, \(2010 ; 125:\) e1270-e1277. \({ }^{9}\) Lefvre, T., et al., "Beer Consumption Increases Human Attractiveness to Malaria Mosquitoes," PLoS ONE, 2010; 5(3): e9546.were assigned to the two groups randomly. The attractiveness to mosquitoes of each volunteer was tested twice: before the beer or water and after. Mosquitoes were released and caught in traps as they approached the volunteers. For the beer group, the total number of mosquitoes caught in the traps before consumption was 434 and the total was 590 after consumption. For the water group, the total was 337 before and 345 after. (a) Define the relevant parameter(s) and state the null and alternative hypotheses for a test to see if, after consumption, the average number of mosquitoes is higher for the volunteers who drank beer. (b) Compute the average number of mosquitoes per volunteer before consumption for each group and compare the results. Are the two sample means different? Do you expect that this difference is just the result of random chance? (c) Compute the average number of mosquitoes per volunteer after consumption for each group and compare the results. Are the two sample means different? Do you expect that this difference is just the result of random chance? (d) If the difference in part (c) is unlikely to happen by random chance, what can we conclude about beer consumption and mosquitoes? (e) If the difference in part (c) is statistically significant, do we have evidence that beer consumption increases mosquito attraction? Why or why not?

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if average sales are higher in stores where customers are approached by salespeople than in stores where they aren't
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