Exam 8: Fitting Probability Models to Frequency Data

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

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

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.

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

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

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.

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

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

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?

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

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

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? ? 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 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?

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

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

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 2 1 16 2 10 \geq3 2

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?

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

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.)

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?