Sample Is Selected From A Population With Μ 80 After A Treat

Question

Sample Is Selected From A Population With Μ 80 After A Treatmen A sample is selected from a population with µ = 80. After a treatment is administered to the individuals, the sample mean is found to be M = 75 and the variance is s2 = 100. a. If the sample has n = 4 scores, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with alpha = 0.05. b. If the sample has n=25 scores, then calculate the estimated standard error and determine whether the sample is sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with alpha = 0.05. c. Describe how increasing the size of the sample affects the standard error and the likelihood of rejecting the null hypothesis.

Dr. Jack HW Helper · Accepted Answer

Introduction In statistical hypothesis testing, understanding how sample size influences the standard error and the decision to reject a null hypothesis is fundamental. This paper explores the effect of sample size on the standard error and the assessment of treatment effects based on a sample mean deviation from a known population mean. We will examine two scenarios involving different sample sizes to illustrate these concepts. Methodology The core approach involves calculating the estimated standard error of the mean (SEM) from sample variance and sample size, then conducting a two-tailed hypothesis test at the 0.05 significance level. The specific formulas used include: Standard Error (SE) = s / √n Test statistic (t) = (M - µ) / SE where s is the sample standard deviation, n the sample size, M the sample mean, and µ the population mean. Results Scenario 1: Sample size n = 4 Given: Population mean, µ = 80 Sample mean, M = 75 Variance, s2 = 100 Sample size, n = 4 First, compute the sample standard deviation: s = √100 = 10 Next, calculate the standard error: SE = 10 / √4 = 10 / 2 = 5 The test statistic t is then: t = (75 - 80) / 5 = -5 / 5 = -1 Using a t-distribution table with df = n - 1 = 3, the critical t-value at α = 0.05 (two-tailed) is approximately 3.182. Since |t| = 1 Scenario 2: Sample size n=25 Given that the variance remains the same: s2 = 100, and now n=25 Standard deviation: s = √100 = 10 Standard error: SE = 10 / √25 = 10 / 5 = 2 Test statistic: t = (75 - 80) / 2 = -5 / 2 = -2.5 Degrees of freedom: df = 24. The critical t-value at α = 0.05 (two-tailed) is approximately 2.064. Since |t| = 2.5 > 2.064, we reject the null hypothesis. This suggests that with n=25, the sample provides sufficient evidence to conclude that the treatment has a significant effect. Discussion: Effect of Sample Size on Standard Error and Hypothesis Testing Increasing the sample size results in a decrease in the standard error because SEM is inversely proportional to the square

Sample Is Selected From A Population With Μ 80 After A Treat ✓ Solved

Sample Is Selected From A Population With Μ 80 After A Treatmen

Sample Paper For Above instruction

Introduction

Methodology

Results

Scenario 1: Sample size n = 4

Scenario 2: Sample size n=25

Discussion: Effect of Sample Size on Standard Error and Hypothesis Testing

Conclusion

References