Einstein claims that his true average quiz score is 90. You have sample data on 25 of his quizzes. The average for that samaple is 83, and the standard deviation for the sample (s) is 4. The alpha for this two-tailed test is 0.05. Provide the calculated t-vaule, the critial t-value, and corresponding confidence interval around the sample average
To calculate the t-value, critical t-value, and confidence interval, we can follow these steps:
Step 1: Calculate the t-value
The formula to calculate the t-value is:
t = (sample average - population average) / (standard deviation / √n)
Here, the sample average is 83, the population average claimed by Einstein is 90, the standard deviation (s) is 4, and the sample size (n) is 25.
t = (83 - 90) / (4 / √25)
t = -7 / (4/5)
t = -7 * 5/4
t = -8.75
Step 2: Calculate the critical t-value
Since the alpha for this two-tailed test is 0.05, we divide this by 2 to get 0.025. With a sample size of 25, we have 24 degrees of freedom (n-1).
Using a t-table or statistical software, we find the critical t-value for a two-tailed test with 0.025 significance level and 24 degrees of freedom to be approximately ±2.064.
Step 3: Calculate the confidence interval
The confidence interval is given by:
CI = sample average ± (critical t-value * standard error)
The standard error (SE) is calculated as:
SE = standard deviation / √n
Substituting the values, we get:
SE = 4 / √25
SE = 4/5
SE = 0.8
CI = 83 ± (2.064 * 0.8)
CI = 83 ± 1.651
CI = (81.349, 84.651)
Therefore, the calculated t-value is -8.75, the critical t-value is ±2.064, and the corresponding confidence interval around the sample average is (81.349, 84.651).