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Determine both Type I and Type II errors for the following scenario:
Assume a null hypothesis, H _{0} , that states the percentage of adults with jobs is at least 88%.
Identify the Type I and Type II errors from these four statements.
Type I error: c
Type I error: b
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters α and β , for a Type I and a Type II error respectively. The power of the test, 1 – β , quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.
α = probability of a Type I error = P (Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.
β = probability of a Type II error = P (Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.
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.
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.
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.
For Exercise 9.12 , what are α and β in words?
α = the probability that you think the bag cannot withstand -15 degrees F, when in fact it can
β = the probability that you think the bag can withstand -15 degrees F, when in fact it cannot
In words, describe 1 – β For Exercise 9.12 .
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.
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.
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?
The power of a test is 0.981. What is the probability of a Type II error?
0.019
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.
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?
0.998
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?
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