Exam 8: Fitting Probability Models to Frequency Data

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? ? critical values for df=2 Sig. level Value 0.05 3.84 0.025 5.02 0.01 6.63

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? ? critical values for df=2 Sig. level Value 0.05 7.81 0.025 9.35 0.01 11.34

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? ? critical values for df=2 Sig. level Value 0.05 5.99 0.025 7.38 0.01 9.21

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? ? Number Number 0 3 1 12 2 9 \geq3 6

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? ? Number Number 0 3 1 12 2 9 \geq3 6

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

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.) ? critical values for df=2 Sig. level Value 0.05 7.81 0.025 9.35 0.01 11.34

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? ? critical values for df=2 Sig. level Value 0.05 7.81 0.025 9.35 0.01 11.34

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

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? ? critical values for df=2 Sig. level Value 0.05 3.84 0.025 5.02 0.01 6.63

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.

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? ? critical values for df=2 Sig. level Value 0.05 5.99 0.025 7.38 0.01 9.21

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? ? Number Number 0 3 1 12 2 9 \geq3 6 critical values for df=2 Sig. level Value 0.05 5.99 0.025 7.38 0.01 9.21 ?

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?

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

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? ? Number Number 0 2 1 16 2 10 \geq3 2

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? ? critical values for df=2 Sig. level Value 0.05 3.84 0.025 5.02 0.01 6.63

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?

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.) ? critical values for df=2 Sig. level Value 0.05 7.81 0.025 9.35 0.01 11.34

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