8.E: Outcomes and the Type I and Type II Errors (Optional Exercises)
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Exercise \(\PageIndex{5}\)
The mean price of mid-sized cars in a region is $32,000. A test is conducted to see if the claim is true. State the Type I and Type II errors in complete sentences.
- Answer
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Type I : The mean price of mid-sized cars is $32,000, but we conclude that it is not $32,000.
Type II : The mean price of mid-sized cars is not $32,000, but we conclude that it is $32,000.
Exercise \(\PageIndex{6}\)
A sleeping bag is tested to withstand temperatures of –15 °F. You think the bag cannot stand temperatures that low. State the Type I and Type II errors in complete sentences.
Exercise \(\PageIndex{7}\)
For Exercise 9.12, what are \(\alpha\) and \(\beta\) in words?
- Answer
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\(\alpha =\) the probability that you think the bag cannot withstand -15 degrees F, when in fact it can
\(\beta =\) the probability that you think the bag can withstand -15 degrees F, when in fact it cannot
Exercise \(\PageIndex{8}\)
In words, describe \(1 - \beta\) For Exercise \(\PageIndex{}\)
Exercise \(\PageIndex{9}\)
A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, \(H_{0}\), is: the surgical procedure will go well. State the Type I and Type II errors in complete sentences.
- Answer
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Type I : The procedure will go well, but the doctors think it will not.
Type II: The procedure will not go well, but the doctors think it will.
Exercise \(\PageIndex{10}\)
A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, \(H_{0}\), is: the surgical procedure will go well. Which is the error with the greater consequence?
Exercise \(\PageIndex{11}\)
The power of a test is 0.981. What is the probability of a Type II error?
- Answer
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0.019
Exercise \(\PageIndex{12}\)
A group of divers is exploring an old sunken ship. Suppose the null hypothesis, \(H_{0}\), is: the sunken ship does not contain buried treasure. State the Type I and Type II errors in complete sentences.
Exercise \(\PageIndex{13}\)
A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, \(H_{0}\), is: the sample does not contain E-coli. The probability that the sample does not contain E-coli, but the microbiologist thinks it does is 0.012. The probability that the sample does contain E-coli, but the microbiologist thinks it does not is 0.002. What is the power of this test?
- Answer
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0.998
Exercise \(\PageIndex{14}\)
A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, \(H_{0}\), is: the sample contains E-coli. Which is the error with the greater consequence?