Spring_2022_Homework_7

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University of California, San Diego *

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MATH 20 F

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Statistics

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Jan 9, 2024

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pdf

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Uploaded by ProfessorLightningDragonfly32

Homework 7 Spring 2022 Due June 3, 2022 @ 10:00pm PST Name: Student ID: Section: Instructions Please read the following instructions carefully: 1. Please show all notation for probability statements. 2. Box your final answers. 3. Please verify that your scans are legible. 4. Please assign pages the the questions when submitting to gradescope. 5. This assignment is due via gradescope on the due date.

Homework 7 1. A manager at a farm is curious about if the size of the Colossal Cantaloupe melons that they grow at the farm has increased. They know that last year, the average weight of the Colossal Cantaloupes that were grown on the farm was 37 pounds. They take a sample of 29 melons and with a mean of 38.4 pounds and a sample standard deviation of 5.1 pounds. Has the weight of the melons increased since last year? Let µ be the true mean weight of the Colossal Cantaloupes grown this year at the farm. Test this claim at the α = 0.05 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under H 0 . Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at the α = 0.05 level and state your conclusion. Page 2

Homework 7 2. The football coach wants to test if a protein shake meal supplement will improve the performance of their team over a 6 week period. 16 football players each drink a protein shake in the morning and in the evening every day for the 6 week period. They each have to complete a 100m sprint both before and after the 6 weeks. The difference in their running speed, in seconds, is computed as SPD AF T − SPD BEF , and the coach wants to see if the protein shakes have made the team faster at running the 100m. The sample mean difference from the 16 students was 0.1 seconds and the sample standard deviations was 0.13 seconds. Perform a test at the α = 0 . 10 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. Page 3

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Homework 7 (d) Draw a picture of the distribution of the test statistic under H 0 . Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at the α = 0.10 level and state your conclusions in context of the problem. Page 4

Homework 7 3. You are a keen gardener and keep a great array of houseplants in your dorm. You are thinking about trying to water them less often because you are worried that you are over-watering,so you carry out an experiment. You randomly choose 12 of your plants and assign them to group 1, you randomly pick another 7 plants and then assign them to group 2. The average growth after 2 weeks for group 1 with the old amount of water is 3cm with a standard deviation of 0.8 cm. For group 2, the group with the new amount of water, the average growth rate is 3.4cm with a standard deviation of 0.45cm. Let µ 1 be the true growth rate under the old amount of watering and µ 2 be the true growth rate under the new amount of watering. Are these two means different? Perform a test at an α = 0 . 01 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under H 0 . Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at the α = 0.01 level and state your conclusions in context of the problem. (f) Construct a 99% confidence interval for the true difference in growth rates from the 2 different fertilisers. (g) Does this interval reaffirm your statistical decision from the hypothesis test? Explain. Page 5

Homework 7 4. A researcher is interested in understanding if there is a difference in the proportion of undergrad and grad students at UCI who prefer online teaching to in person teaching, at the α = 0 . 05 level. They take 2 samples, first, a sample of 300 undergrad students. The second, is a sample of 172 grad students. Of the undergrads, 186 said they preferred online lectures, and of the graduate students, 104 said that they prefer online lectures. Let p 1 = the proportion of undergrad students who prefer online class and p 2 = the proportion of grad students who prefer online lectures. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under H 0 . Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at the α = 0.05 level and state your conclusions in the context of the problem. Page 6

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Homework 7 5. According to the LA Times, ”In an annual survey of more than 500 beaches, Heal the Bay reported ... that 92% of the state’s (California) beaches had logged good water-quality marks between April and Octo- ber of 2019”. ( https://www.latimes.com/california/story/2020-06-30/california-beach-report-card ). We want to know if measures to increase the beach water quality in California that have been put in place since 2019 are working, that is, if the percentage of beaches with good water-quality marks has increased since 2019. Let p be the true proportion of beaches that deserve good-quality water marks this year. To test if the measures of beach improvement are working, you want to see if more than 92% of beaches get good marks at an α = 0 . 05 level. A sample of 368 beaches is taken in January 2022 and of these beaches 347 logged a good water-quality rating. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under H 0 . Label and provide values for the critical value and the test statistic, and shade the critical region. Page 7

Homework 7 (e) Make and justify a statistical decision at the α = 0.05 level and state your conclusions in context of the problem. (f) If we increased the sample size to be very large ( > 1000) but kept the sample proportion the same how (if at all) would you expect your decision of this hypothesis test to change? Page 8

Homework 7 6. For the multiple choice questions, circle the best answer. In hypothesis testing, a Type 1 error occurs when: A. the null hypothesis is not rejected when the alternative hypothesis is true. B. the null hypothesis is rejected when the alternative hypothesis is true. C. the null hypothesis is not rejected when the null hypothesis is true. D. the null hypothesis is rejected when the null hypothesis is true. In hypothesis testing, a Type 2 error occurs when: A. the null hypothesis is not rejected when the alternative hypothesis is true. B. the null hypothesis is not rejected when the null hypothesis is true. C. the null hypothesis is rejected when the alternative hypothesis is true. D. the null hypothesis is rejected when the null hypothesis is true. In a hypothesis test, if the null hypothesis is actually false, what type of error could be made? A. Type 1. B. Type 2. C. Type 1 if it’s a one-sided test and Type 2 if it’s a two-sided test. D. Type 2 if it’s a one-sided test and Type 1 if it’s a two-sided test. If we decide to reject the null hypothesis, which type of mistake could have been made when making this decision? A. Type 1. B. Type 2. C. Type 1 if it’s a one-sided test and Type 2 if it’s a two-sided test. D. Type 2 if it’s a one-sided test and Type 1 if it’s a two-sided test. In an American criminal trial, the null hypothesis is that the defendant is innocent and the alternative hypothesis is that the defendant is guilty. Which of the following describes a Type 2 error for a criminal trial? A. A not guilty verdict for a person who is innocent B. A guilty verdict for a person who is innocent. C. A guilty verdict for a person who is not innocent. D. A not guilty verdict for a person who is guilty A decrease in α in a statistical test for parameter θ , would make which of the following true? (select all that apply) A. Type I error decreases B. Type II error increases C. Power decreases D. V ar ( θ ) decreases The power of a statistical test increases when: (select all that apply) A. sample size is increased B. lowering α C. the value of the true parameter is closer to the null D. test sensitivity is decreased Page 9

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Homework 7 If I performed a test with α = 0.05, what is the probability that I correctly fail to reject the null hypothesis? A. 0.05 B. 0.10 C. 0.95 D. Not enough information to answer the question. Page 10

Spring_2022_Homework_7

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