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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

State the null and alternative hypotheses for the statistical test described. Testing to see if there is evidence that a proportion is greater than 0.3 .

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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?

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 to see if a well-known company is lying in its advertising. If there is evidence that the company is lying, the Federal Trade Commission will file a lawsuit against them.
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