Use the following formulas to return the t-scores you need in the above formula from Excel. =t.inv(probability, degrees of freedom) To retrieve the lower t (t. part of formula): probability = a/2 and degrees of freedom = n-1 To retrieve the upper t (t part of formula): probability = (1-a/2) and degrees of freedom = n-1 Confidence interval for a sample proportion: ptmargin of error [p(1-p) pH* Similarly, we can deconstruct this formula as well. p+z* [[P(1-P)] Using this version of the formula, if I retrieve a z-value = -2, I would directly plug that value in as z: Lower bound: p + (-2)*, where ZH is the t-score for the upper bound (z-high) Upper bound: p + Z* where z, is the t-score for the lower bound (z-low) where zi is the t-score for the upper bound (z-high) Use the following formula to return the z-scores you need in the above formula from Excel. =norm.s.inv(probability) To retrieve the lower z (z. part of formula): probability = a/2 To retrieve the upper z (z part of formula): probability = 1 -a/2 A sample of 41 triathlon participants required an average time of 105 minutes to complete the race with a standard deviation of 20 minutes. 1. Start constructing the 90% confidence interval for the population mean. What are t, and th? a = 1-confidence level = 1-0.9 -0.1 a/2 = 0.05 Using Excel find the lower t-score: t.inv(0.05, 40) = -1.6839 Note: Excel will give you very precise t-scores, I will round these at the fourth decimal place. 2. Start constructing the 90% confidence interval for the population mean. What is the margin of error? Hint: use this formula margin of error = t *(-) 3. What is the 90% confidence interval for the population mean? 4. What is the 95% confidence interval for the population mean? 5. Is the interval range for the 95% confidence interval larger or smaller than the 90% confidence interval? In a sample of a large number of teaching union members, 90% intend to vote for the new labor contract. 6. What is the minimum sample size needed to meet the condition: n * p > 5? 7. What is the minimum sample size needed to meet the condition: 12* (1-p) 5? 8. If n = 50, what is the 95% confidence interval? 9. If n = 500, what is the 95% confidence interval? 10. Is the interval range for the 95% confidence interval larger or smaller with n=500 compared to with n=50? Formulas Confidence interval for a sample mean: * + margin of error x t, * ( We can deconstruct this formula to get the lower and upper bound values separately. where th is the t-score for the upper bound (t-high) Note: since I am suggesting below that you retrieve both the lower and upper t-scores (this practice will help us in future work), it may be easier to think of the lower bound formula using this version of the equation where you can directly plug-in the negative t-value you retrieve from Excel: Lower bound = x + + where t, is the t-score for the lower bound (t-low) For example, if I retrieve a t-value = -2, I would directly plug that value in as tu: Lower bound = x + (-2)*() where th is the t-score for the upper bound (t-high) Upper bound = x + t * | We then represent it as an interval range = [lower bound, upper bound]