1 Chapter 7, Using Excel: Confidence Intervals These pages demonstrate the Excel functions that can be used to calculate confidence intervals. • Chapter 7.2 - Estimating a Population Mean (σ known) 2 Here, Excel can calculate the critical value (zα/2) and/or the margin of error (E) defined by σ E = z p : α/2 n This uses the NORM.S.INV and/or the CONFIDENCE.NORM functions. • Chapter 7.3 - Estimating a Population Proportion 3 Here, Excel can calculate the critical value (zα/2) used in the margin of error defined by s p^ q^ E = z : α/2 n This uses the NORM.S.INV function. You must then complete the calculations to get the margin of error (E). • Chapter 7.4 - Estimating a Population Mean (σ unknown) 4 Here, Excel can calculate the critical value (tα/2) and/or the margin of error (E) defined by s E = t p : α/2 n This uses the T.INV and/or the CONFIDENCE.T functions. 2 Chapter 7.2 - Estimating a Population Mean (σ known) Here, Excel can calculate the critical value (zα/2) and/or the margin of error (E) defined by σ E = z p ; α/2 n • Notation: { E = the margin of error { zα/2 = the critical value of z. { σ is the population standard deviation { n is the sample size { α = 1 − confidence level (in decimal form) ∗ If the confidence level is 90% then α = 1 − :90 = 0:10. ∗ If the confidence level is 95% then α = 1 − :95 = 0:05. ∗ If the confidence level is 99% then α = 1 − :99 = 0:005. • Finding the critical value zα/2 Here we use the NORM.S.INV function. NORM.S.INV stands for the inverse of the standard normal distribution (z-distribution). General Usage: NORM.S.INV(area to the left of the critical value) Specific Usage: zα/2 = NORM.S.INV (1 − α=2) Example: If you want zα/2 for a 95% confidence interval, use zα/2 = NORM.S.INV(0.975) = 1.960 • Finding the margin of error E Here we use the CONFIDENCE.NORM function. CONFIDENCE.NORM stands for the confidence interval from a normal distribution. Usage: CONFIDENCE.NORM(α, σ, n) Example: If you want a 95% confidence interval for a mean when the population standard deviation is 10.2 from a sample of size 35, the margin of error would be E = CONFIDENCE.NORM(0.05, 10.2, 35) = 3.3792 3 Chapter 7.3 - Estimating a Population Proportion Here, Excel can calculate the critical value (zα/2) used in the margin of error defined by s p^ q^ E = z : α/2 n However, you must then complete the calculations to get the margin of error (E). • Notation: { E = the margin of error { zα/2 = the critical value of z. { p^ is the sample proportion andq ^ = 1 − p^. { n is the sample size { α = 1 − confidence level (in decimal form) ∗ If the confidence level is 90% then α = 1 − :90 = 0:10. ∗ If the confidence level is 95% then α = 1 − :95 = 0:05. ∗ If the confidence level is 99% then α = 1 − :99 = 0:005. • Finding the critical value zα/2 Here we use the NORM.S.INV function. NORM.S.INV stands for the inverse of the standard normal distribution (z-distribution). General Usage: NORM.S.INV(area to the left of the critical value) Specific Usage: zα/2 = NORM.S.INV (1 − α=2) Example: If you want zα/2 for a 95% confidence interval, use zα/2 = NORM.S.INV(0.975) = 1.960 4 Chapter 7.4 - Estimating a Population Mean (σ unknown) Here, Excel can calculate the critical value (tα/2) and/or the margin of error (E) defined by s E = t p α/2 n • Notation: { E = the margin of error { tα/2 = the critical value of z. { s is the sample standard deviation { n is the sample size { α = 1 − confidence level (in decimal form) ∗ If the confidence level is 90% then α = 1 − :90 = 0:10. ∗ If the confidence level is 95% then α = 1 − :95 = 0:05. ∗ If the confidence level is 99% then α = 1 − :99 = 0:005. • Finding the critical value tα/2 Here we use the T.INV function. T.INV stands for the inverse of the t-distribution. General Usage: T.INV(area left of critical value, degrees of freedom) Specific Usage: tα/2 = T.INV (1-α=2, df) Example: If you want tα/2 for a 95% confidence interval based in a sample of size 20, use tα/2 = T.INV(0.975, 19) = 2.093 • Finding the margin of error E Here we use the CONFIDENCE.T function. CONFIDENCE.T stands for the confidence interval from a t-distribution. Usage: CONFIDENCE.T(α, s, n) Example: If you want a 95% confidence interval for a mean with a sample standard deviation of 10.2 from a sample of size 35, the margin of error would be E = CONFIDENCE.T(0.05, 10.2, 35) = 3.5038.