Statistical Inference.WA unit4

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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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1Written Assignment Unit 4University of the PeopleMATH 1281-01: Statistical InferenceInstructor: Dr. Hitesh VermaDecember 11, 2024

2Part 1A survey of high school seniors was conducted by the National Center of Education Statistics.They were collecting test data on reading, writing, and several other subjects. A random sample of 250 students was examined from the survey. Side-by-side boxplots of reading and writing scores as well as a histogram of the differences in scores are shown below. (a) HypothesesThe research question asks if there is a significant difference in the average scores of studentsin the reading and writing exams.Null Hypothesis (H)₀: There is no significant difference in the average scores of reading and writing exams.Alternative Hypothesis (Hₐ): There is a significant difference in the average scores of reading and writing exams.This is a two-tailed test since we are looking for any difference, not just one direction (Diez et al., 2019).(b) Checking ConditionsTo perform the t-test, the following conditions should be checked:Paired Data

3The data is paired because the scores come from the same students for reading and writing exams.Random SampleThe sample of 250 students was randomly selected.Normality of DifferencesThe histogram of differences in scores should be approximately symmetric without significant skewness or outliers (Heisinger & Hoyle, 2012).(c) Performing the T-TestThe test statistic for paired data is calculated as:t = (x - x ) / (s / √n)ₐₑWhere:xₐ = -0.545 (mean difference in scores)s = 8.887 (standard deviation of differences)n = 250 (number of paired observations)Step 1:Calculate the standard error (SE)SE = s / √nSE = 8.887 / √250 ≈ 8.887 / 15.8114 ≈ 0.562Step 2:Calculate tt = -0.545 / 0.562 ≈ -0.97

4Step 3:P-Value and Conclusion The p-value is given as . Since (at a 5% significance level), we fail to reject the null hypothesis.Conclusion:There is not enough evidence to conclude a significant difference in the average scores of reading and writing exams (Diez et al., 2019).(d) Type of ErrorIf we fail to reject the null hypothesis when it is actually false, we would be making a Type IIerror. A Type II error means that we did not detect a difference in the average scores of reading and writing exams when one actually exists (Diez et al., 2019)..(e) Confidence Interval and 0A confidence interval includes all plausible values for the population parameter. Since we failed to reject the null hypothesis, the confidence interval for the average difference is likely to include 0.The p-value of 0.39 suggests that the observed difference is not statistically significant, meaning 0 is a plausible value for the true mean difference.

5Part 2Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions. Does this data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage? Assume that conditions for inference are satisfied. (1) State the HypothesisWe aim to test whether there is a significant difference in the average city mileage between cars with manual and automatic transmissions. The hypotheses are:Null Hypothesis (H): There is no difference in the average city mileage between cars with ₀manual and automatic transmissions. μ_manual = μ_automatic.Alternative Hypothesis (H): There is a difference in the average city mileage ₐbetween cars with manual and automatic transmissions. μ_manual ≠ μ_automatic.(2) Calculate the T-StatisticThe formula for the t-statistic is:t = (x₁ - x₂) / √((s₁² / n₁) + (s₂² / n₂))Where:x = 19.85 (mean for manual transmissions)₁

6x = 16.12 (mean for automatic transmissions)₂s= 4.51 (standard deviation for manual transmissions)₁s= 3.58 (standard deviation for automatic transmissions)₂n= 26 (sample size for manual transmissions)₁n= 26 (sample size for automatic transmissions)₂Step-by-step calculation:1. Compute the variances:s₁² / n₁ = (4.51²) / 26 = 20.3401 / 26 ≈ 0.7823s₂² / n₂ = (3.58²) / 26 = 12.8164 / 26 ≈ 0.49372. Add the variances:0.7823 + 0.4937 = 1.2763. Take the square root:√1.276 ≈ 1.13014. Compute the t-statistic:t = (19.85 - 16.12) / 1.1301 = 3.73 / 1.1301 ≈ 3.30(3) Calculate the Degrees of FreedomThe formula for degrees of freedom (df) is:df = ((s² / n) + (s² / n))² / (((s² / n)² / (n- 1)) + ((s² / n)² / (n- 1)))₁₁₂₂₁₁₁₂₂₂Step-by-step calculation:

71. Compute the numerator:((s² / n) + (s² / n))² = (0.7823 + 0.4937)² = 1.276² = 1.628₁₁₂₂2. Compute the denominator:(s₁² / n₁)² / (n₁ - 1) = (0.7823²) / 25 = 0.61199529 / 25 ≈ 0.02448(s₂² / n₂)² / (n₂ - 1) = (0.4937²) / 25 = 0.24374269 / 25 ≈ 0.009750.02448 + 0.00975 = 0.034233. Compute the degrees of freedom:df = 1.628 / 0.03423 ≈ 47.56 ≈ 48(4) ConclusionGiven the t-statistic = 3.30 and a p-value = 0.0029, which is less than the significance level (α= 0.05), we reject the null hypothesis.There is strong evidence to suggest that the average city mileage differs significantly betweencars with manual and automatic transmissions.

8ReferencesDiez, D., Cetinkaya-Rundel, M., & Barr, C. D. (2019). OpenIntro Statistics-Fourth Edition. Open Textbook Library. https://www.biostat.jhsph.edu/~iruczins/teaching/books/2019.openintro.statistics.pdf