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Chapter 7: Q4E (page 397)

What is the difference between Type I and Type II errors in hypothesis testing? How do α and β relate to Type I and Type II errors?

Short Answer

Expert verified

Type I error is the rejection of the true null hypothesis, while Type II error occurs when one fails to reject the false null hypothesis. The value of Type I error is predefined and thus can be controlled, whereas Type II error can not be controlled.

Step by step solution

01

Given Information

In a hypothesis testing problem taking the right decision about the null hypothesis is very important. Otherwise, it may lead to the errors known as Type I and Type II errors.

02

Stating the difference between the two types of errors

Type I error is the rejection of the true null hypothesis, while Type II error occurs when one fails to reject the false null hypothesis. The value of Type I error is predefined and thus can be controlled, whereas Type II error can not be controlled.

A Type I error is denoted by α , and a Type II error β . A Type I error is also known as the size of the test.

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Most popular questions from this chapter

Question: Testing the placebo effect. The placebo effect describes the phenomenon of improvement in the condition of a patient taking a placebo—a pill that looks and tastes real but contains no medically active chemicals. Physicians at a clinic in La Jolla, California, gave what they thought were drugs to 7,000 asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, 70% of the patients reported an improved condition. Use this information to test (at α = 0.05) the placebo effect at the clinic. Assume that if the placebo is ineffective, the probability of a patient’s condition improving is 0.5.

Radon exposure in Egyptian tombs. Refer to the Radiation Protection Dosimetry (December 2010) study of radon exposure in Egyptian tombs, Exercise 6.30 (p. 349). The radon levels—measured in becquerels per cubic meter (\({{Bq} \mathord{\left/ {\vphantom {{Bq} {{m^3}}}} \right. \\} {{m^3}}}\) )—in the inner chambers of a sample of 12 tombs are listed in the table shown below. For the safety of the guards and visitors, the Egypt Tourism Authority (ETA) will temporarily close the tombs if the true mean level of radon exposure in the tombs rises to 6,000\({{Bq} \mathord{\left/ {\vphantom {{Bq} {{m^3}}}} \right. \\} {{m^3}}}\) . Consequently, the ETA wants to conduct a test to determine if the true mean level of radon exposure in the tombs is less than 6,000\({{Bq} \mathord{\left/ {\vphantom {{Bq} {{m^3}}}} \right. \\} {{m^3}}}\) , using a Type I error probability of .10. An SPSS analysis of the data is shown at the bottom of the page. Specify all the elements of the test: \({H_0}\,,{H_a}\) test statistic, p-value,\(\alpha \) , and your conclusion.

50 910 180 580 7800 4000 390 12100 3400 1300 11900 110

Intrusion detection systems. The Journal of Research of the National Institute of Standards and Technology (November– December 2003) published a study of a computer intrusion detection system (IDS). The IDS is designed to provide an alarm whenever unauthorized access (e.g., an intrusion) to a computer system occurs. The probability of the system giving a false alarm (i.e., providing a warning when no intrusion occurs) is defined by the symbol α, while the probability of a missed detection (i.e., no warning given when an intrusion occurs) is defined by the symbol β. These symbols are used to represent Type I and Type II error rates, respectively, in a hypothesis-testing scenario

a. What is the null hypothesis, H0?

b. What is the alternative hypothesis,Ha?

c. According to actual data collected by the Massachusetts Institute of Technology Lincoln Laboratory, only 1 in 1,000 computer sessions with no intrusions resulted in a false alarm. For the same system, the laboratory found that only 500 of 1,000 intrusions were actually detected. Use this information to estimate the values of αand β.

Accuracy of price scanners at Walmart. Refer to Exercise 6.129 (p. 377) and the study of the accuracy of checkout scanners at Walmart stores in California. Recall that the National Institute for Standards and Technology (NIST) mandates that for every 100 items scanned through the electronic checkout scanner at a retail store, no more than two should have an inaccurate price. A study of random items purchased at California Walmart stores found that 8.3% had the wrong price (Tampa Tribune, Nov. 22, 2005). Assume that the study included 1,000 randomly selected items.

a. Identify the population parameter of interest in the study.

b. Set up H0 and Ha for a test to determine if the true proportion of items scanned at California Walmart stores exceeds the 2% NIST standard.

c. Find the test statistic and rejection region (at a=0.05 ) for the test.

d. Give a practical interpretation of the test.

e. What conditions are required for the inference, part d, to be valid? Are these conditions met?

Jury trial outcomes. Sometimes, the outcome of a jury trial defies the “common sense” expectations of the general public (e.g., the 1995 O. J. Simpson verdict and the 2011 Casey Anthony verdict). Such a verdict is more acceptable if we understand that the jury trial of an accused murderer is analogous to the statistical hypothesis-testing process. The null hypothesis in a jury trial is that the accused is innocent. (The status-quo hypothesis in the U.S. system of justice is innocence, which is assumed to be true until proven beyond a reasonable doubt.) The alternative hypothesis is guilt, which is accepted only when sufficient evidence exists to establish its truth. If the vote of the jury is unanimous in favor of guilt, the null hypothesis of innocence is rejected, and the court concludes that the accused murderer is guilty. Any vote other than a unanimous one for guilt results in a “not guilty” verdict. The court never accepts the null hypothesis; that is, the court never declares the accused “innocent.” A “not guilty” verdict (as in the Casey Anthony case) implies that the court could not find the defendant guilty beyond a reasonable doubt

a. Define Type I and Type II errors in a murder trial.

b. Which of the two errors is the more serious? Explain.

c. The court does not, in general, know the values of α and β ; but ideally, both should be small. One of these probabilities is assumed to be smaller than the other in a jury trial. Which one, and why?

d. The court system relies on the belief that the value of is made very small by requiring a unanimous vote before guilt is concluded. Explain why this is so.

e. For a jury prejudiced against a guilty verdict as the trial begins, will the value ofα increase or decrease? Explain.

f. For a jury prejudiced against a guilty verdict as the trial begins, will the value of β increase or decrease? Explain
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