Econ+15B+--+6+--+Final+--+March+16%2C+2022 (1)

.docx

School

Columbia University *

*We aren’t endorsed by this school

Course

15B

Subject

Economics

Date

May 31, 2024

Type

docx

Pages

Uploaded by DeaconHare3679

NAME: _________________________________ I.D.: _____________________________________ FINAL PROBABILITY AND STATISTICS, ECON 15B MARCH 16, 2022 1. For the following seven (7) experiments/questions, pick the most appropriate statistical test. You have the following statistical tests as choices: some may be used more than once, others not at all. Assume homogeneity of variance (where applicable) and the validity of parametric tests (where applicable), unless something is directly stated (e.g., “the data are not at all normal”) or otherwise indicated (viz., by the inspection of the data) which would indicate a strong and obvious violation of an assumption. This means you must inspect the data for violations of all assumptions. Unless it is stated that the population parameter is known, assume it isn’t. Please do not concern yourself with any intervening variable that you may perceive. Finally, please don’t think about what the data would look like in reality. Assume that the question represents reality. One sample z-test used to determine whether the mean of a single sample differs significantly from a known population mean when the population standard deviation is known. One sample t- test used to determine whether the mean of a single sample differs significantly from a known population mean. However, when the population standard deviation is unknown and must be estimated from the sample. t- test for difference between means for two related samples used to determine whether the means of two related samples differ significantly. It is typically used when the samples are related in some way, such as before and after measurements or matched pairs. t-test for the difference between means for two independent samples with homogeneity of variance used to determine whether the means of two independent samples differ significantly. It assumes that the variances of the two populations are equal. t-test for the difference between means for two independent samples with heterogeneity of variance used to determine whether the means of two independent samples differ significantly. However, it does NOT assume equal variances between the populations. a one sample z-test for proportions used to determine whether the proportion of a single sample differs significantly from a known population proportion. It is commonly used in hypothesis testing for categorical data. chi-square goodness of fit used to determine whether the distribution of categorical data differs significantly from a hypothesized distribution. It is often used when dealing with categorical variables with more than two categories . two-sample z-test for the difference between proportions used to determine whether the difference between proportions in two independent samples differs significantly. It is commonly used in comparing proportions between two groups. chi-square test of independence used to determine whether there is a significant association between two categorical variables . It is often used to assess relationships between variables in contingency tables . simple regression used to model the relationship between a single independent variable and a dependent variable . It helps to understand how changes in the independent variable are associated with changes in the dependent variable. multiple regression used to model the relationship between multiple independent variables and a dependent variable . It allows for the examination of the combined association of several predictors on the outcome variable. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above An experimenter wants to conduct a test on whether people who work in cubicles have more office friendships than people who work in private offices. The experimenter takes a random sample of people from a large company and records whether they work in a cubicle or private office and how many co-workers they have had non-work-related conversations with in the past year. Cubicle workers: 6 26 29 23 5 32 9 3 34 8 Private office workers: 3 19 18 22 3 5 16 2 17 4

two-sample z-test : for the difference between proportions tests whether the difference in the proportions of office friendships between people who work in cubicles and people who work in private offices is statistically significant. H0: There is no difference in the proportion of office friendships between people who work in cubicles and people who work in private offices. HA: There is a difference in the proportion of office friendships between people who work in cubicles and people who work in private offices. Chi-square test of independence examines whether there is a statistically significant relationship between two categorical variables: the type of workspace (cubicle or private office) and the frequency of office friendships. H0: There is no association between workspace type and the frequency of office friendships. HA: There is an association between workspace type and the frequency of office friendships. 2. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above Dr. Smith was a great believer in blood pressure as a key indicator of overall health. Her new assistant, Nurse Ben, believed that aa person’s height and weight affect a person’s blood pressure. Nurse Ben wanted to see if this is true. Multiple Regression: Nurse Ben can examine the coefficients associated with height and weight to determine their individual effects on blood pressure, while also assessing the overall fit of the model to the data. Additionally, multiple regression provides information about the strength and direction of the relationships between variables and can help identify potential confounding factors that may influence the relationship between height, weight, and blood pressure. 3. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above A scientist has discovered a cave of 7 dinosaur skulls and wants to know if they belong to the rare Zotosaurus family. On average, the Zotosaurus has a 470-inch-long skull. The dinosaur skulls in the cave are 395 inches on average. One sample t-test: A one sample t-test is appropriate in this scenario because the scientist wants to compare the average skull length of the dinosaur skulls found in the cave (395 inches) with the known average skull length of the Zotosaurus family (470 inches). The one sample t-test will assess whether the observed difference in mean skull length between the sample and the known population mean is statistically significant, helping the scientist determine if the skulls found in the cave likely belong to the rare Zotosaurus family. Why not z-test ? In a one-sample z-test, you typically use the population standard deviation, σ, if it is known. This test is suitable when the population standard deviation is known or when the sample size is large enough to approximate the population standard deviation. When the sample size is small and the population standard deviation is unknown, the appropriate test to use is a one-sample t-test. 4. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above Does the presence of more species of plants increase the productivity of a natural area, as measured by the total mass of plant material? The number of plant species was recorded as well as the productivity. Simple regression would be the appropriate test in this scenario because it allows us to assess the relationship between two continuous variables: the number of plant species (independent variable) and the productivity of the natural area (dependent variable).

5. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above While traveling through South America, a student decides to analyze which country is the friendliest. He measures friendliness by seeing whether or not a person, when asked for directions, gives a reply. For each country, he gathers data on hundreds of people. The four countries analyzed are Argentina, Chile, Brazil, and Uruguay. A chi-square test of independence is appropriate in this scenario because the student wants to analyze whether there is an association between the country a person is from (Argentina, Chile, Brazil, Uruguay) and their likelihood of giving a reply when asked for directions. This test will determine whether there is a significant relationship between the two categorical variables: country and likelihood of giving a reply. H0: There is no association between the country a person is from and their likelihood of giving a reply when asked for directions. HA: There is an association between the country a person is from and their likelihood of giving a reply when asked for directions. 6. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above Zoe takes a sample of eight UCI students to see if test anxiety exists. She measures their pulse a week before a test and then again 10 minutes before the test. The data are as follows: Week Before pulse: 65 68 90 78 59 80 80 76 72 10 Minutes Before pulse: 70 75 96 87 65 87 87 84 80 The paired samples t-test : determine whether there is a significant difference between the mean pulse rates of the students measured a week before the test and those measured 10 minutes before the test. This test is appropriate because it accounts for the paired nature of the data, allowing Zoe to assess whether there is a statistically significant change in pulse rate before and shortly before a test, which could indicate the presence of test anxiety. H0: There is no significant difference between the mean pulse rates of UCI students measured a week before the test and those measured 10 minutes before the test. (μ1 = μ2) HA: There is a significant difference between the mean pulse rates of UCI students measured a week before the test and those measured 10 minutes before the test. (μ1 ≠ μ2) 7. Please simply write the letter for the test as your answer. Here are the tests: A: one sample z-test B: one-sample t-test C: t-test for the difference between means for two related samples D: t-test for the difference between means for two independent samples with homogeneity of variance E: t-test for the difference between means for two independent samples with heterogeneity of variance F: a one sample z-test for proportions (or a chi-square goodness of fit) G: chi-square goodness of fit only (where a one sample z-test of proportions isn’t appropriate) H: a two-sample z-test for the difference between proportions (or a chi-square test of independence) I: chi-square test of independence only (where a two-sample z-test for the difference between proportions isn’t appropriate.) J: simple regression K : multiple regression L: none of the above The UCI Study Abroad Center is interested in whether UCI students are different from US college students in where they study abroad. They thus take a sample of 500 UCI students. It is known that among US college students, 60% study abroad in Europe, 20% in Asia & Oceania, 10% in the Americas, and 10% in the Middle East and Africa. Among the sample of UCI students, 60% study abroad in Europe, 33% in Asia & Oceania, 5% in the Americas and 2% in the Middle East and Africa. Two-sample z-test: for the difference between proportions would be appropriate in this scenario because the UCI Study Abroad Center wants to compare the proportions of UCI students studying abroad in different regions with the proportions of US college students studying abroad in the same regions. H0: There is no difference between the proportions of UCI students studying abroad in each region and the proportions of US college students studying abroad in each region. HA: There is a difference between the proportions of UCI students studying abroad in each region and the proportions of US college students studying abroad in each region.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version