Outcomes and the Type I and Type II Errors (Exercises)

Last updated
Save as PDF

Page ID: 5076

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

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

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

$\alpha =$ 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

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: 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: 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?

Search

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)

Exercise \(\PageIndex{11}\)

Exercise \(\PageIndex{12}\)

Exercise \(\PageIndex{13}\)

Exercise \(\PageIndex{14}\)