ICA9-Spring 2024-NEW DATA-KEY

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North Carolina State University *

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311

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Statistics

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Apr 3, 2024

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ST 311 Week 9 In-Class Activity 1 | P a g e Hypothesis Tests for Proportions Problem 1: For each statement in parts A and B, do the following: • Write a hypothesis pair using correct notation • Label which hypothesis is the claim • State the direction of the test as left-tail, right-tail, or two-tail • State a decision in terms of the null hypothesis, either “Reject 𝐻 0 ” or “Fail to reject 𝐻 0 ” • Write a sentence interpreting the decision in terms of the original claim. A) The proportion of drivers who admit to running red lights is less than 0.35 . The P−value is 0.091 and the level of significance is 0.1. 𝐻 0 : ? = 0.35 𝐻 𝐴 : ? < 0.35 Left-tailed test Since p-value (0.091) < significance level ( 𝛼 = 0.10 ), we Reject 𝑯 𝟎 . There is enough evidence at the 10% significance level to suggest that the proportion of drivers who admit to running red lights is less than 0.35. B) A majority of young adults spend more than 3.5 hours per day using a smartphone. The P−value is 0.41 and the level of significance is 0.05. 𝐻 0 : ? = 0.50 𝐻 𝐴 : ? > 0.50 (majority is more than half) Right-tailed test Since p-value (0.41) > significance level ( 𝛼 = 0.05 ), we Fail to Reject 𝑯 𝟎 . There is not enough evidence at the 5% significance level to suggest that a majority of young adults spend more than 3.5 hours per day using a smartphone.

ST 311 Week 9 In-Class Activity 2 | P a g e Problem 2: LGBTQIA+ community organizers want to know if people who identify as LGBTQIA+ ( read more here ) are proportionally represented at NC State University undergraduate programs compared to the overall population of the state of North Carolina. In North Carolina, roughly 4% of the population identifies as LGBTQIA+ ( Williams Institute ), and NC State University currently has 26,329 undergraduate students enrolled ( NC State University ). Students who are enrolled in ST 311 are selected as a sample for the study. A total of 628 students completed the survey, and 20 students identified as members of the LGBTQIA+ group. An additional 33 students opened the survey but did not respond to the survey item. The organizers want to determine if this data can serve as evidence of the underrepresentation of LGBTQIA+ students at NC State University compared to the state population. Conduct an appropriate hypothesis test using a significance level of 5%. A) What are the hypotheses? 𝐻 0 : ? = 0.04 𝐻 𝐴 : ? < 0.04 (i.e. underrepresented) B) What is the sample statistic? ?̂ = 20 628 = 0.0318 C) What is the null distribution of the test statistic? What do you have to check to determine this? If ?? ≥ 10 ??? ?? ≥ 10 , and the sample is random, then the null distribution will follow a normal. ?? = 25.12, ?? = 602.88 D) What is the test statistic for this hypothesis test? 𝑧 ∗ = ?̂ − ? √ ?? ? = 0.0318 − 0.04 √ 0.04 ∗ 0.96 628 = −1.05 E) What is the p-value? Interpret this value. ? = 𝑃(? < −1.05) = 0.1486 The probability that a random sample of 628 NCSU students has a sample proportion of LGBTQIA+ as small or smaller than 0.0318, if the true proportion of LGBTQIA+ in NC is 0.04, is 0.1486.

ST 311 Week 9 In-Class Activity 3 | P a g e F) What decision should be made? Since p = 0.1486 is not considered unusually or significantly low (since it is greater than 𝛼 = 0.05 ), we Fail to Reject 𝐻 0 . G) Write an interpretation of your decision. There is not sufficient evidence at the 5% significance level to suggest that the LGBTQIA+ population is underrepresented by undergraduates at NCSU. H) Could selecting ST 311 courses as a sample for the entire undergraduate population undermine the outcome of this study? Why or why not? Answers may vary, but should include a discussion of random sampling techniques and possible bias.

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ST 311 Week 9 In-Class Activity 4 | P a g e You can get involved in supporting LGBTQIA+ communities at NC State University: https://diversity.ncsu.edu/pridecenter/ Problem 3: In a Gallup poll of 1018 randomly selected adults, 110 said that cloning of humans should be allowed. Test the claim that more than 10% of all adults say that cloning of humans should be allowed. A) Write the hypotheses for this test. 𝐻 0 : ? = 0.10 𝐻 𝐴 : ? > 0.10 B) Check all conditions. The sample is random. ?? = 101.8, ?? = 916.2 C) Calculate the standardized test statistic. How can we interpret this value? ?̂ = 110 1018 = 0.1081 𝑧 ∗ = ?̂ − ? √ ?? ? = 0.1081 − 0.10 √ 0.10 ∗ 0.90 1018 = 0.861 (?? 0.857 𝑖? ?𝑥???) The sample proportion (110 out of 1018) who think cloning of humans should be allowed lies 0.861 standard deviations above the assumed proportion of 10%. (This is not unusual, it’s within 1 SD) D) Without using StatCrunch, which value below is your best estimate of the p-value for this test? Draw a picture that represents the p-value as a guide. o ? = 0.09 o ? = 0.10 o ? = 0 .20 o ? = 0.40 o ? = 0.80 o ? = 0.90 E) At the 10% significance level, what decision is appropriate? Since the p-value is about 20% (0.20), this is very high and not at all unusual in the null distribution (if p really is 10%). We should Fail to Reject 𝐻 0 . F) Would your decision change if you conducted a two-sided test for this data? No – a two-sided test ( 𝐻 0 : ? = 0.10 𝑣? 𝐻 𝐴 : ? ≠ 0.10 ) would result in a p-value twice as large as the one above!

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