University:
University of the People
Course:
MATH 1281 | Statistical Inference
Academic year:

2024
Views:

38

Pages:

5
Author:

amrinderjeet kaur

UNIT 2 LEARNING JOURNAL Part 1 1a. Hypothesis Test for Proportion Step 1: Define Hypotheses - Null Hypothesis (H₀): p = 0.50 (50% of adults who do not attend college cite affordability as the reason) - Alternative Hypothesis (Hᵠ): p < 0.50 (Less than 50% of adults cite affordability as the reason) Step 2: Check Conditions - Independence Condition: The sample of 441 adults is randomly selected, and since it is significantly smaller than the total adult population, the observations are independent. - Success-Failure Condition: Expected counts under the null hypothesis are: Successes: np₀ = 441(0.50) = 220.5 Failures: n(1 - p₀) = 441(0.50) = 220.5 Since both values are greater than 10, the normal approximation holds. Step 3: Compute Test Statistic Sample proportion (ș) = 0.38, Population proportion (p₀) = 0.50, Sample size (n) = 441 Formula for the test statistic: Step 4: Compute P-value Using the standard normal table, P(Z < -5.06) is approximately 0.0000. Step 5: Conclusion Since the p-value is significantly lower than the typical alpha level (0.05), we reject the null hypothesis. There is strong evidence that less than 50% of adults cite affordability as the reason for not attending college. 1b. Sample Size Calculation for 90% Confidence Level Formula for margin of error (ME): Given: ● ME = 0.015 ● Confidence level = 90%, z* = 1.645 ● p = 0.38 Rearranging the formula to solve for n: Thus, a survey of at least 2,832 people is needed. Part 2 2a. Confidence Interval for the Difference Between Proportions Proportion of sleep-deprived individuals: ● Texas: p₁ = 0.07, n₁ = 13,270 ● Dallas: p₂ = 0.068, n₂ = 4,681 Step 1: Validate Conditions 1. Independence Condition: Samples are randomly selected and much smaller than the population. 2. Success-Failure Condition: ○ Texas: n₁p₁ = 929, n₁(1 - p₁) = 12,341 ○ Dallas: n₂p₂ = 318, n₂(1 - p₂) = 4,363 Since all values exceed 10, conditions are met. Step 2: Compute Confidence Interval Formula: Since the confidence interval includes 0, there is no statistically significant difference between sleep deprivation rates. 2b. Hypothesis Test for Proportion Difference Step 1: Define Hypotheses ● H₀: p₁ - p₂ = 0 (No difference in sleep deprivation rates) ● Hᵠ: p₁ - p₂ \neq 0 (A difference exists) Step 2: Compute Test Statistic Pooled proportion: Standard error: Step 3: Compute P-value Using the normal table, P(|Z| > 0.463) = 0.6434. Step 4: Conclusion Since p = 0.6434 > 0.05, we fail to reject H₀. There is insufficient evidence to suggest a significant difference in sleep deprivation rates between Texas and Dallas residents. REFERENCES: Moore, D. S., McCabe, G. P., Craig, B. A., & Alwan, L. C. (2021). The practice of statistics for business and economics (5th ed.). W. H. Freeman.