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Chapter 4: Problem 133

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing a new drug with potentially dangerous side effects to see if it is significantly better than the drug currently in use. If it is found to be more effective, it will be prescribed to millions of people.

Short Answer

Expert verified
It is more appropriate to use a relatively small significance level, such as \(\alpha = 0.01\), to minimize the risk of falsely declaring the new drug as more effective.

Step by step solution

01

Understanding the context

In this situation, we are testing a new drug with potentially dangerous side effects to know if it is significantly better than the currently used drug. If we find it more effective, it will be prescribed to millions of people.
02

Interpreting significance level

A significance level (\(\alpha\)) is the probability of rejecting the null hypothesis when it is actually true. Therefore, a larger \(\alpha\) means a greater chance of wrongly rejecting the null hypothesis (Type I error). In the context of this experiment, it would mean a greater chance of concluding that the new drug is more effective when it actually isn't.
03

Conclusion

Considering the stakes, including dangerous side effects and the drug potentially being prescribed to millions of people, a Type I error could have serious consequences. Consequently, it is better to use a relatively small \(\alpha\) (such as \(\alpha = 0.01\)). The lower \(\alpha\) reduces the chance of falsely concluding that the new drug is more effective, therefore minimizing the risk of prescribing an ineffective or potentially harmful drug to millions of people.

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Key Concepts

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

Type I Error
In hypothesis testing, a Type I error occurs when we reject a true null hypothesis. This means we believe our alternative idea or theory is correct, even if it's not. It's like saying a new treatment works better than an old one, when in reality, it doesn鈥檛. This can create serious issues, especially in fields like medicine.
  • A Type I error can lead to wrong decisions being made.
  • Choosing a smaller significance level (\( \alpha \)) can help minimize this risk.
If our goal is to ensure public safety, like when testing a drug, we must be cautious about making Type I errors. Using a lower \( \alpha \), such as 0.01, helps reduce the likelihood of making this mistake, ensuring more confidence in our test results.
Drug Testing
When testing a new drug, scientists must determine if it outperforms existing treatments without causing serious side effects. This is a critical decision-making process which can affect many people.
  • Drugs must be both effective and safe for widespread use.
  • Researchers examine whether new treatments work better than old ones.
To do this effectively, clear and reliable evidence is needed. In the case of a drug that might have dangerous side effects, testing requires careful consideration to avoid Type I errors. Using a more stringent significance level like 0.01 ensures that any claim about the drug's effectiveness is backed by solid evidence. This reduces the risk of endorsing a faulty medication.
Hypothesis Testing
Hypothesis testing is a process used to decide if there is enough evidence to reject a given assumption or null hypothesis about a population parameter. It's a step-by-step procedure that helps researchers prove or disprove theories.
  • First, a null hypothesis assumes no effect or difference exists.
  • The alternative hypothesis suggests a significant effect or difference.
The significance level (\( \alpha \)) is central to hypothesis testing. It sets the threshold for how confident we must be to reject the null hypothesis. A smaller \( \alpha \) indicates stronger evidence is needed to make a claim against the null hypothesis. For something critical, like drug testing with potential health impacts, setting a lower \( \alpha \) is necessary. This precaution helps avoid claiming benefits of a new drug when it's not justified, thereby protecting public health.

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Most popular questions from this chapter

Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

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 there is evidence that a correlation between height and salary is significant (that is, different than zero ).

Mating Choice and Offspring Fitness Does the ability to choose a mate improve offspring fitness in fruit flies? Researchers have studied this by taking female fruit flies and randomly dividing them into two groups; one group is put into a cage with a large number of males and able to freely choose who to mate with, while flies in the other group are each put into individual vials, each with only one male, giving no choice in who to mate with. Females are then put into egg laying chambers, and a certain number of larvae collected. Do the larvae from the mate choice group exhibit higher survival rates? A study \(^{44}\) published in Nature found that mate choice does increase offspring fitness in fruit flies (with p-value \(<0.02\) ), yet this result went against conventional wisdom in genetics and was quite controversial. Researchers attempted to replicate this result with a series of related experiments, \({ }^{45}\) with data provided in MateChoice. (a) In the first replication experiment, using the same species of fruit fly as the original Nature study, 6067 of the 10000 larvae from the mate choice group survived and 5976 of the 10000 larvae from the no mate choice group survived. Calculate the p-value. (b) Using a significance level of \(\alpha=0.05\) and \(\mathrm{p}\) -value from (a), state the conclusion in context. (c) Actually, the 10,000 larvae in each group came from a series of 50 different runs of the experiment, with 200 larvae in each group for each run. The researchers believe that conditions dif- fer from run to run, and thus it makes sense to treat each \(\mathrm{run}\) as a case (rather than each fly). In this analysis, we are looking at paired data, and the response variable would be the difference in the number of larvae surviving between the choice group and the no choice group, for each of the 50 runs. The counts (Choice and NoChoice and difference (Choice \(-\) NoChoice) in number of surviving larva are stored in MateChoice. Using the single variable of differences, calculate the p-value for testing whether the average difference is greater than \(0 .\) (Hint: this is a single quantitative variable, so the corresponding test would be for a single mean.) (d) Using a significance level of \(\alpha=0.05\) and the p-value from (c), state the conclusion in context. (e) The experiment being tested in parts (a)-(d) was designed to mimic the experiment from the original study, yet the original study yielded significant results while this study did not. If mate choice really does improve offspring fitness in fruit flies, did the follow-up study being analyzed in parts (a)-(d) make a Type I, Type II, or no error? (f) If mate choice really does not improve offspring fitness in fruit flies, did the original Nature study make a Type I, Type II, or no error?

It is well established that exercise is beneficial for our bodies. Recent studies appear to indicate that exercise can also do wonders for our brains, or, at least, the brains of mice. In a randomized experiment, one group of mice was given access to a running wheel while a second group of mice was kept sedentary. According to an article describing the study, "The brains of mice and rats that were allowed to run on wheels pulsed with vigorous, newly born neurons, and those animals then breezed through mazes and other tests of rodent IQ"9 compared to the sedentary mice. Studies are examining the reasons for these beneficial effects of exercise on rodent (and perhaps human) intelligence. High levels of BMP (bonemorphogenetic protein) in the brain seem to make stem cells less active, which makes the brain slower and less nimble. Exercise seems to reduce the level of BMP in the brain. Additionally, exercise increases a brain protein called noggin, which improves the brain's ability. Indeed, large doses of noggin turned mice into "little mouse geniuses," according to Dr. Kessler, one of the lead authors of the study. While research is ongoing in determining how strong the effects are, all evidence points to the fact that exercise is good for the brain. Several tests involving these studies are described. In each case, define the relevant parameters and state the null and alternative hypotheses. (a) Testing to see if there is evidence that mice allowed to exercise have lower levels of BMP in the brain on average than sedentary mice. (b) Testing to see if there is evidence that mice allowed to exercise have higher levels of noggin in the brain on average than sedentary mice. (c) Testing to see if there is evidence of a negative correlation between the level of BMP and the level of noggin in the brains of mice.

Could owning a cat as a child be related to mental illness later in life? Toxoplasmosis is a disease transmitted primarily through contact with cat feces, and has recently been linked with schizophrenia and other mental illnesses. Also, people infected with Toxoplasmosis tend to like cats more and are 2.5 times more likely to get in a car accident, due to delayed reaction times. The CDC estimates that about \(22.5 \%\) of Americans are infected with Toxoplasmosis (most have no symptoms), and this prevalence can be as high as \(95 \%\) in other parts of the world. A study \(^{37}\) randomly selected 262 people registered with the National Alliance for the Mentally Ill (NAMI), almost all of whom had schizophrenia, and for each person selected, chose two people from families without mental illness who were the same age, sex, and socioeconomic status as the person selected from NAMI. Each participant was asked whether or not they owned a cat as a child. The results showed that 136 of the 262 people in the mentally ill group had owned a cat, while 220 of the 522 people in the not mentally ill group had owned a cat. (a) This is known as a case-control study, where cases are selected as people with a specific disease or trait, and controls are chosen to be people without the disease or trait being studied. Both cases and controls are then asked about some variable from their past being studied as a potential risk factor. This is particularly useful for studying rare diseases (such as schizophrenia), because the design ensures a sufficient sample size of people with the disease. Can casecontrol studies such as this be used to infer a causal relationship between the hypothesized risk factor (e.g., cat ownership) and the disease (e.g., schizophrenia)? Why or why not? (b) In case-control studies, controls are usually chosen to be similar to the cases. For example, in this study each control was chosen to be the same age, sex, and socioeconomic status as the corresponding case. Why choose controls who are similar to the cases? (c) For this study, calculate the relevant difference in proportions; proportion of cases (those with schizophrenia) who owned a cat as a child minus proportion of controls (no mental illness) who owned a cat as a child. (d) For testing the hypothesis that the proportion of cat owners is higher in the schizophrenic group than the control group, use technology to generate a randomization distribution and calculate the p-value. (e) Do you think this provides evidence that there is an association between owning a cat as a child and developing schizophrenia? \(^{38}\) Why or why not?
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