Deck 8: Fitting Probability Models to Frequency Data

Question 1

Which of the following best describes how an χ² goodness-of-fit test is used and performed?

A) The observed and expected frequencies are compared and a close match results in a large P-value, which supports rejection of the null hypothesis.
B) The observed and expected frequencies are compared and a close match results in a small P-value, which supports rejection of the null hypothesis.
C) The observed and expected frequencies are compared and a mismatch results in a large P-value, which supports rejection of the null hypothesis.
D) The observed and expected frequencies are compared and a mismatch results in a small P-value, which supports rejection of the null hypothesis.

Answer 1

D

Question 2

Which of the following best describes how the concept of degrees of freedom (df) is used in an χ² goodness-of-fit test?

A) The df describes the strength of our conclusion.
B) The df specifies which probability distribution we will use.
C) The df is based on a list of how many hypotheses we are testing.
D) The df is inversely related to the sample size.

Answer 2

B

Question 3

Consider a situation in which we expect one-third of the observed values to be in each of three categories. We can use an χ² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 14, 19, and 27, what is the χ² value we would obtain?

A) 2.05
B) 2.13
C) 4.30
D) 4.39

Answer 3

C

Question 4

Consider a situation in which we expect one-third of the observed values to be in each of three categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 14, 19, and 27, what is the P-value range we would obtain for our test? (Use the table of critical values shown to answer this question.)
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 5

Consider a situation in which we expect one-third of the observed values to be in each of 3 categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 14, 19, and 27, and using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}$

A) Fail to reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
B) Fail to reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.
C) Reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
D) Reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.

Question 6

Consider a situation in which we expect one third of the observed values to be in each of three categories. We can use a χ² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 13, 18, and 29, what is the χ² value we would obtain?

A) 6.28
B) 6.42
C) 6.56
D) 6.70

Question 7

Consider a situation in which we expect one-third of the observed values to be in each of three categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 13, 18, and 29, what is the P-value range we would obtain for our test? (Use the table of critical values shown to answer this question.)
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 8

Consider a situation in which we expect one-third of the observed values to be in each of three categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 13, 18, and 29, and using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}$

A) Fail to reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
B) Fail to reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.
C) Reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
D) Reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.

Question 9

Consider a situation in which we expect the same frequency of 64 total observations to be in each of four categories. We can use a χ² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 12, 24, 10, and 18, what is the χ² value we would obtain?

A) 7.50
B) 7.82
C) 9.50
D) 9.82

Question 10

Consider a situation in which we expect the same frequency of 64 total observations to be in each of four categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 12, 24, 10, and 18, what is the P-value range we would obtain for our test? (Use the table of critical values shown to answer this question.)
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 7.81\\0.025 & 9.35 \\0.01 & 11.34\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 11

Consider a situation in which we expect the same frequency of 64 total observations to be in each of four categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 12, 24, 10, and 18, and using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 7.81\\0.025 & 9.35 \\0.01 & 11.34\end{array}\end{array}$

A) Fail to reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
B) Fail to reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.
C) Reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
D) Reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.

Question 12

Consider a situation in which we expect the same frequency of 64 total observations to be in each of four categories. We can use an χ² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 12, 26, 8, and 18, what is the χ² value we would obtain?

A) 10.55
B) 11.50
C) 12.45
D) 13.40

Question 13

Consider a situation in which we expect the same frequency of 64 total observations to be in each of four categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 12, 26, 8, and 18, what is the P-value range we would obtain for our test? (Use the table of critical values shown to answer this question.)
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 7.81\\0.025 & 9.35 \\0.01 & 11.34\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 14

Consider a situation in which we expect the same frequency of 64 total observations to be in each of four categories. We can use an ?² goodness-of-fit test to test whether the frequencies of offspring are as expected. If the numbers of values in each category are 12, 26, 8, and 18, and using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 7.81\\0.025 & 9.35 \\0.01 & 11.34\end{array}\end{array}$

A) Fail to reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
B) Fail to reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.
C) Reject the null hypothesis, we lack the evidence to decide that the frequencies differ from those that were expected.
D) Reject the null hypothesis, we have good evidence to decide that the frequencies differ from those that were expected.

Question 15

When two heterozygotes are mated, the ratios of the offspring produced should be in a 1:2:1 ratio if normal Mendelian segregation is occurring. If one of the alleles is dominant, then the phenotypes observed should be present in a 3:1 ratio with the dominant phenotype more common than the recessive one. We can use an χ² goodness-of-fit test to test whether the ratio of offspring is indeed 3:1. Imagine a cross is performed and the number of offspring observed are 65 with the dominant phenotype and 35 with the recessive phenotype. What is the χ² value we would obtain?

A) 2.267
B) 3.000
C) 4.396
D) 5.333

Question 16

When two heterozygotes are mated, the ratios of the offspring produced should be in a 1:2:1 ratio if normal Mendelian segregation is occurring. If one of the alleles is dominant, then the phenotypes observed should be present in a 3:1 ratio with the dominant phenotype more common than the recessive one. We can use an ?² goodness-of-fit test to test whether the ratio of offspring is indeed 3:1. Imagine a cross is performed and the number of offspring observed are 65 with the dominant phenotype and 35 with the recessive phenotype. Using the table of critical values shown, what is the P-value range we would obtain for our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 17

When two heterozygotes are mated, the ratios of the offspring produced should be in a 1:2:1 ratio if normal Mendelian segregation is occurring. If one of the alleles is dominant, then the phenotypes observed should be present in a 3:1 ratio with the dominant phenotype more common than the recessive one. We can use an ?² goodness-of-fit test to test whether the ratio of offspring is indeed 3:1. Imagine a cross is performed and the number of offspring observed are 65 with the dominant phenotype and 35 with the recessive phenotype. Using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) We fail to reject the null hypothesis and therefore conclude that something other than usual Mendelian segregation is occurring.
B) We fail to reject the null hypothesis and therefore conclude that usual Mendelian segregation is occurring.
C) We reject the null hypothesis and therefore conclude that something other than usual Mendelian segregation is occurring.
D) We reject the null hypothesis and therefore conclude that usual Mendelian segregation is occurring.

Question 18

When two heterozygotes are mated, the ratios of the offspring produced should be in a 1:2:1 ratio if normal Mendelian segregation is occurring. If one of the alleles is dominant, then the phenotypes observed should be present in a 3:1 ratio with the dominant phenotype more common than the recessive one. We can use a χ² goodness-of-fit test to test whether the ratio of offspring is indeed 3:1. Imagine a cross is performed and the number of offspring observed are 67 with the dominant phenotype and 33 with the recessive phenotype. What is the χ² value we would obtain?

A) 2.895
B) 3.413
C) 3.864
D) 4.273

Question 19

When two heterozygotes are mated, the ratios of the offspring produced should be in a 1:2:1 ratio if normal Mendelian segregation is occurring. If one of the alleles is dominant, then the phenotypes observed should be present in a 3:1 ratio with the dominant phenotype more common than the recessive one. We can use an ?² goodness-of-fit test to test whether the ratio of offspring is indeed 3:1. Imagine a cross is performed and the number of offspring observed are 67 with the dominant phenotype and 33 with the recessive phenotype. Using the table of critical values shown, what is the P-value range we would obtain for our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 20

When two heterozygotes are mated, the ratios of the offspring produced should be in a 1:2:1 ratio if normal Mendelian segregation is occurring. If one of the alleles is dominant, then the phenotypes observed should be present in a 3:1 ratio with the dominant phenotype more common than the recessive one. We can use an ?² goodness-of-fit test to test whether the ratio of offspring is indeed 3:1. Imagine a cross is performed and the number of offspring observed are 67 with the dominant phenotype and 33 with the recessive phenotype. Using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) We fail to reject the null hypothesis and therefore conclude that something other than usual Mendelian segregation is occurring.
B) We fail to reject the null hypothesis and therefore conclude that usual Mendelian segregation is occurring.
C) We reject the null hypothesis and therefore conclude that something other than usual Mendelian segregation is occurring.
D) We reject the null hypothesis and therefore conclude that usual Mendelian segregation is occurring.

Question 21

Consider a claim that 60% of the mice in a region have parasitic infections. We can use an χ² goodness-of-fit test to test whether the proportion is indeed 60%. Consider a study in which we collect a random sample of 50 mice and 37 have infections. What is the χ² value we would obtain?

A) 1.500
B) 1.883
C) 4.083
D) 5.094

Question 22

Consider a claim that 60% of the mice in a region have parasitic infections. We can use an ?² goodness-of-fit test to test whether the proportion is indeed 60%. Consider a study in which we collect a random sample of 50 mice and 37 have infections. Calculate the ?² value, and using the table of critical values shown, what is the P-value range we would obtain for our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) 0.05 < P
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 23

Consider a claim that 60% of the mice in a region have parasitic infections. We can use an ?² goodness-of-fit test to test whether the proportion is indeed 60%. Consider a study in which we collect a random sample of 50 mice and 37 have infections. Calculate the ?² value, and using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) We fail to reject the null hypothesis and therefore conclude that the percentage of mice with infections may indeed be 60% as claimed.
B) We fail to reject the null hypothesis and therefore conclude that the percentage of mice with infections is not 60% as claimed.
C) We reject the null hypothesis and therefore conclude that the percentage of mice with infections may indeed be 60% as claimed.
D) We reject the null hypothesis and therefore conclude that the percentage of mice with infections is not 60% as claimed.

Question 24

Consider a claim that 60% of the mice in a region have parasitic infections. We can use an χ² goodness-of-fit test to test whether the proportion is indeed 60%. Consider a study in which we collect a random sample of 50 mice and 36 have infections. What is the χ² value we would obtain?

A) 1.500
B) 3.000
C) 4.500
D) 6.000

Question 25

Consider a claim that 60% of the mice in a region have parasitic infections. We can use an ?² goodness-of-fit test to test whether the proportion is indeed 60%. Consider a study in which we collect a random sample of 50 mice and 36 have infections. Calculate the ?² value, and using the table of critical values shown, what is the P-value range we would obtain for our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) P > 0.05
B) 0.025 < P < 0.05
C) 0.01 < P < 0.025
D) P < 0.01

Question 26

Consider a claim that 60% of the mice in a region have parasitic infections. We can use an ?² goodness-of-fit test to test whether the proportion is indeed 60%. Consider a study in which we collect a random sample of 50 mice and 36 have infections. Calculate the ?² value, and using the table of critical values shown, what is the conclusion of our test?
?
$\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 3.84 \\0.025 & 5.02 \\0.01 & 6.63\end{array}\end{array}$

A) We fail to reject the null hypothesis and therefore conclude that the percentage of mice with infections may indeed be 60% as claimed.
B) We fail to reject the null hypothesis and therefore conclude that the percentage of mice with infections is not 60% as claimed.
C) We reject the null hypothesis and therefore conclude that the percentage of mice with infections may indeed be 60% as claimed.
D) We reject the null hypothesis and therefore conclude that the percentage of mice with infections is not 60% as claimed.

Question 27

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) What is the mean number of birds observed on each light?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 3 \\1 & 12 \\2 & 9 \\\geq 3 & 6 \\\hline\end{array}$

A) 1.40
B) 1.60
C) 1.80
D) 2.00

Question 28

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) What is the variance for the number of birds observed on each light?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 3 \\1 & 12 \\2 & 9 \\\geq 3 & 6 \\\hline\end{array}$

A) 0.87
B) 1.27
C) 1.67
D) 2.07

Question 29

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) Assuming a Poisson process, what is the expected number of streetlights without any birds?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 3 \\1 & 12 \\2 & 9 \\\geq 3 & 6 \\\hline\end{array}$

A) 3.06
B) 4.06
C) 5.06
D) 6.06

Question 30

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) Assuming a Poisson process, what is the expected number of streetlights with three or more birds?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 3 \\1 & 12 \\2 & 9 \\\geq 3 & 6 \\\hline\end{array}$

A) 5.50
B) 6.50
C) 7.50
D) 8.50

Question 31

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) What is the ?² value we would obtain for a goodness-of-fit test comparing these data to the expectations from a Poisson process?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 3 \\1 & 12 \\2 & 9 \\\geq 3 & 6 \\\hline\end{array}$

A) 2.33
B) 4.33
C) 6.33
D) 8.33

Question 32

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) Using the ?² value we obtained for a goodness-of-fit test comparing these data to the expectations from a Poisson process and the list of critical values shown, what is the conclusion of our test?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 3 \\1 & 12 \\2 & 9 \\\geq 3 & 6 \\\hline\end{array}$ $\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}$ ?

A) Fail to reject the null hypothesis, the birds appear randomly distributed on the streetlights
B) Fail to reject the null hypothesis, the birds appear non-randomly distributed on the streetlights
C) Reject the null hypothesis, the birds appear randomly distributed on the streetlights
D) Reject the null hypothesis, the birds appear non-randomly distributed on the streetlights

Question 33

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) What is the mean number of birds observed on each light?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 2 \\1 & 16 \\2 & 10 \\\geq 3 & 2 \\\hline\end{array}$

A) 1.40
B) 1.60
C) 1.80
D) 2.00

Question 34

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) What is the variance for the number of birds observed on each light?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 2 \\1 & 16 \\2 & 10 \\\geq 3 & 2 \\\hline\end{array}$

A) 0.32
B) 0.52
C) 0.72
D) 0.92

Question 35

Consider a study testing whether birds were equally likely to rest on each streetlight. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) Assuming a Poisson process, what is the expected number of streetlights without any birds?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 2 \\1 & 16 \\2 & 10 \\\geq 3 & 2 \\\hline\end{array}$

A) 3.4
B) 5.4
C) 7.4
D) 9.4

Question 36

Consider a study testing whether the number of birds resting on streetlights is random. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) Assuming a Poisson process, what is the expected number of streetlights with three or more birds?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 2 \\1 & 16 \\2 & 10 \\\geq 3 & 2 \\\hline\end{array}$

A) 5.00
B) 6.00
C) 7.00
D) 8.00

Question 37

Consider a study testing whether the number of birds resting on streetlights is random. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) What is the ?² value we would obtain for a goodness-of-fit test comparing these data to the expectations from a Poisson process?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 2 \\1 & 16 \\2 & 10 \\\geq 3 & 2 \\\hline\end{array}$

A) 3.85
B) 5.85
C) 7.85
D) 9.85

Question 38

Consider a study testing whether the number of birds resting on streetlights is random. Researchers surveyed 30 randomly chosen streetlights in a city known to have many birds and counted the number of birds resting on each streetlight. Data on the number of birds seen on the lights are shown. (Note: No lights had more than three birds.) Using the ?² value we obtained for a goodness-of-fit test comparing these data to the expectations from a Poisson process and the list of critical values shown, what is the conclusion of our test?
?
$\begin{array}{|cc|}\hline \begin{array}{l}\text { Number } \\\underline {\text { birds }}\end{array} & \begin{array}{l}\text { Number } \\\underline {\text { of lights }}\end{array} \\0 & 2 \\1 & 16 \\2 & 10 \\\geq 3 & 2 \\\hline\end{array}$ $\begin{array}{c}\chi^{2} \text { critical values }\\\text { for } d f=2\\\begin{array}{ll}\text { Sig. level } & \text { Value } \\\hline 0.05 & 5.99 \\0.025 & 7.38 \\0.01 & 9.21\end{array}\end{array}$ ?

A) Fail to reject the null hypothesis, the birds appear randomly distributed on the streetlights.
B) Fail to reject the null hypothesis, the birds appear non-randomly distributed on the streetlights.
C) Reject the null hypothesis, the birds appear randomly distributed on the streetlights.
D) Reject the null hypothesis, the birds appear non-randomly distributed on the streetlights.

Question 39

At the end of the chapter, there was a list of guidelines and procedures for conducting a good statistical study. Which of the following was not one of the broad procedures listed?

A) Complete the ethical permission paperwork accurately.
B) Discuss the design with other researchers.
C) Keep the design of your experiment as simple as possible.
D) List the possible outcomes of your experiment.

Question 40

A χ² goodness-of-fit test that results in a large P-value supports the null hypothesis.

Question 41

A χ² goodness-of-fit test directly compares frequencies, not proportions.

Question 42

A χ² goodness-of-fit test can be done when the expected value in one or more categories is less than 1 as long as long as at least one category has an expected value larger than 5.

Question 43

A χ² goodness-of-fit test can be done when the expected value in each category is more than 1.

Question 44

The χ² goodness-of-fit test is a good alternative for the binomial test when there are only two categories and a large number of observations.

Question 45

The χ² goodness-of-fit test makes a better estimate of the true P-value than the binomial test.

Question 46

The P-value obtained using an χ² goodness-of-fit test is a good indication of the magnitude of the difference in observed and expected proportions.

Question 47

If a set of values exhibits a Poisson distribution, then the mean of the values is the same as the variance of the values.

Question 48

If a set of values exhibits a Poisson distribution with a mean value of 1, then the probability of observing no successes in a given period is equal to 0.368. (Note: The value of e = 2.718.)

Question 49

Describe how we use critical values to estimate the probability of seeing the χ² test statistic we calculate.

Question 50

Anthony says that he thinks births at a local hospital will be equally likely to occur on any day of the week, whereas Justin says they probably won't be because doctors or hospitals schedule Caesarian sections for certain days. Imagine they collect data for 105 births and the numbers on each day are as follow: 21, 16, 8, 12, 10, 17, 21. Conduct an χ² goodness-of-fit test to determine whether the data support Anthony or Sarah. As part of your answer, present the test statistic and the P-value range it corresponds to (using the table of critical values for 6 degrees of freedom shown).
​

Question 51

Sofia says that she thinks births at a local hospital will be equally likely to occur on any day of the week, whereas Isabella says they probably won't be because doctors or hospitals schedule Caesarian sections for certain days. Imagine they collect data for 105 births and the numbers on each day are as follow: 20, 18, 7, 12, 9, 17, 22. Conduct an χ² goodness-of-fit test to determine whether the data support Sofia or Isabella. As part of your answer, present the test statistic and the P-value range it corresponds to (using the table of critical values for 6 degrees of freedom shown).
​
​

Question 52

Consider a Poisson distribution with an integer mean value. Using the equation for the Poisson distribution, show that the probability value for the number of observations equal to the mean and the probability value for a number of observations one less than the mean are equal.

Question 53

Outline the major steps you would take when planning an experiment testing whether a high-fat diet increases obesity in rats. Keep in mind the advice at the end of the chapter (the interleaf) when you describe the steps.