Exam 8: Fitting Probability Models to Frequency Data

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.

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? ? Number Number 0 2 1 16 2 10 \geq3 2 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.) What is the mean number of birds observed on each light? ? Number Number 0 3 1 12 2 9 \geq3 6

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). ​ critical values for df=2 Sig. level Value 0.05 12.59 0.025 14.45 0.01 16.81

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?

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

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.

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?

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

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

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

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

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?