Statistical Inference.WA unit2

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University of the People**We aren't endorsed by this school

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CS 1281

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Mathematics

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Dec 19, 2024

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1Hypothesis Testing and Confidence Intervals for ProportionsUniversity of the PeopleMATH 1281-01: Statistical InferenceInstructor: Dr. Hitesh VermaNovember 28, 2024

2Part 1A group of 441 adults who did not have a college degree and were not currently enrolled in school were randomly selected. 38% of them said they did not attend college because they could not afford it.a.Conduct a hypothesis test to determine if there is strong evidence supporting the statement that less than 50% of adults who decide not to attend college are because they cannot afford it. State the hypotheses and validate the independence and success-failure condition. Compute test statistics, and p-value, interpret the data, and conclude if the null hypothesis needs to be rejected or not.Step 1: State the HypothesesNull hypothesis (H): p = 0.50 ₀Alternative hypothesis (Hₐ): p < 0.50 Step 2: Verify Conditions1.Independence condition: The sample is random, and n = 441 is less than 10% of the population. 2.Success-Failure condition: n * p̂ = 441 * 0.38 = 167.58 n * (1 - p̂) = 441 * 0.62 = 273.42 Both values are greater than 10 (Diez et al., 2019).

3Step 3: Compute Test StatisticFormula: z = (p̂ - p₀) / √(p₀ * (1 - p₀) / n)Substitute values:z = (0.38 - 0.50) / √(0.50 * 0.50 / 441) z = (-0.12) / √(0.25 / 441) z = (-0.12) / 0.0237 ≈ -5.06Step 4: Compute p-valueFor z = -5.06, the p-value is approximately p = 0.00001.Step 5: ConclusionSince p < 0.05, reject H. There is strong evidence that less than 50% of adults cite ₀affordability as a reason for not attending college.b. Suppose we wanted the margin of error for the 90% confidence level to be about 1.5%. How large of a survey would you recommend? Formula: n = (z² * p̂ * (1 - p̂)) / E²Substitute values:z = 1.645 (for 90% confidence)E = 0.015 (margin of error)p̂ = 0.38n = (1.645)² * 0.38 * 0.62 / (0.015)²n = (2.706 * 0.2356) / 0.000225n ≈ 2825.36Sample size needed:n = 2826 (rounded up).

4Part 2A random sample study was conducted on 13,270 Texas and 4,681 Dallas residents. It was found that the proportion of residents who reported insufficient rest or sleep during each of the preceding 31 days is 7.0% in Texas and 6.8% in Dallas. a.Calculate a 95% confidence interval for the difference between the proportions of sleep-deprived individuals among Texas residents and Dallas residents. Explain the validation of independence and success-failure conditions. Construct the interval and interpret it in the context of this study. Step 1: Validate Conditions 1. Independence condition: Random sampling from large populations.2. Success-Failure condition: Texas: n * p̂ = 13,270 * 0.07 = 929.9, n * (1 - p̂) = 12,340.1 Dallas: n * p̂ = 4,681 * 0.068 = 318.3, n * (1 - p̂) = 4,362.7 Both conditions are satisfied (Diez et al., 2019).Step 2: Confidence Interval FormulaFormula: (p̂₁ - p̂₂) ± z * √((p̂₁ * (1 - p̂₁)) / n₁ + (p̂₂ * (1 - p̂₂)) / n₂)Substitute: p̂= 0.07, p̂= 0.068, n= 13,270, n= 4,681, z = 1.96₁₂₁₂CI = (0.07 - 0.068) ± 1.96 * √((0.07 * 0.93) / 13,270 + (0.068 * 0.932) / 4,681)CI = 0.002 ± 0.0078CI = (-0.0058, 0.0098)

5Interpretation:The difference in proportions is between -0.0058 and 0.0098, suggesting no significant difference.b.Conduct a hypothesis test to determine if the provided data is strong evidence for the rate of sleep deprivation is different for the two states given α = 0.05. Calculate the test statistics, and p-value and provide a conclusion to support your observation.Step 1: State Hypotheses(H_0: p_1 = p_2\) (H_a: p_1 \neq p_2\)Step 2: Test Statistic Formula z = (p̂₁ - p̂₂) / √[ p̂(1 - p̂) * (1/n₁ + 1/n₂) ]Where p̂ = (x+ x) / (n+ n):₁₂₁₂p̂ = (929 + 318) / (13,270 + 4,681) ≈ 0.0698Substitute:z = (0.07 - 0.068) / √[ 0.0698 * (1 - 0.0698) * (1/13,270 + 1/4,681) ] ≈ 0.36Step 3: p-value For z = 0.36, p ≈ 0.719Step 4: Conclusion

6Since p > 0.05* fail to reject H. There is no significant evidence that the sleep ₀deprivation rates differ between Texas and Dallas residents.References Diez, D. M., Barr, C. D., & Çetinkaya-Rundel, M. (2019). Openintro statistics - Fourth edition.Open Textbook Library. https://www.biostat.jhsph.edu/~iruczins/teaching/books/2019.openintro.statistics.pdfThe Organic Chemistry Tutor. (2019a, October 28). Hypothesis testing - solving problems with proportions[Video]. YouTube.