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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 √ . 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 pˆ qˆ 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 √ . 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 √ , 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 pˆ qˆ 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. – pˆ is the sample proportion andq ˆ = 1 − pˆ. – 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 √ 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