This is one of the following five articles on Confidence Intervals in Excel

z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013

t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013

Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean

Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013

Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

# Min Sample Size to Limit

Width of a Confidence

Interval of a Population

Proportion

The larger the sample taken, the smaller the Confidence Interval becomes. That makes intuitive sense because the more sample information that is gathered, the more tightly the position of the population mean can be defined. The Confidence Interval is an interval believed to contain the population mean with specific degree of certainty.

As sample size increase, the Confidence Interval shrinks because greater certainty has been attained. The margin of error, which is equal to half the width of the Confidence Interval, therefore shrinks as well.

During the design phase of a statistical experiment, sample size should be determined. Sampling has a cost and additional sampling beyond what is necessary to attain a desired level of certainty is often undesirable. One common objective of the design phase of a statistical test involving sampling is to determine the minimum sample size required to obtain a specified degree of certainty.

This minimum sample size, n, can be derived by the following equation:

(Half-width of C.I.) = z Value_{α, 2-tailed} * SQRT[ (p_{est} * q_{est}) / n]

Estimates of population parameters p and q must be used in this equation because sample statistics p_bar and q_bar are not available since a sample has not been taken.

(Half-width of C.I.) = z Value_{α, 2-tailed} * SQRT[ (p_{est} * q_{est}) / n]

(Half-width of C.I.) = NORM.S.INV(1 – α/2) * SQRT[ (p_{est} * q_{est}) / n]

Squaring both sides gives the following:

(Half-width of C.I.)^{2} = NORM.S.INV^{2} (1 – α/2) * p_{est} * q_{est} / n

Further algebraic manipulation provides the following:

n = [NORM.S.INV^{2} (1 – α/2) * p_{est} * q_{est}] / (Half-width of C.I.)^{2}

or, equivalently

n = [NORM.S.INV^{2} (1 – α/2) * p_{est} * q_{est}] / (Margin of Error)^{2}

The count of data observations in a sample, n, must be a whole number so n must be rounded up to the nearest whole number. This is implemented in Excel as follows:

n = Roundup**(** **(** NORM.S.INV^{2} (1 – α/2) * p_{est} * q_{est} **)** / **(**Half-width of C.I.**)**^{2} **)**

p_{est} and q_{est} are estimates of the actual population parameters p and q. The most conservative estimate of the minimum sample size would use p_{est} = 0.50.

If p_{est} = 0.05, then q_{est} = 1 – p = 0.50

The product p_{est} * q_{est} has its maximum value of 0.25 when p_{est} = 0.50. This maximum value of p_{est} * q_{est} produces the highest and therefore most conservative value of the minimum sample size, n.

If p is fairly close to 0.5, then p_{est} should be set at 0.5. If p is estimated to be significantly different than 0.5, p_{est} should be set to its estimated value.

### Example 1 of Calculating Min Sample Size in Excel

**Min Number of Voters Surveyed to Limit Poll Error Margin**

Two candidates are running against each other in a national election. This election is considered fairly even. What is the minimum number of voters who should be randomly surveyed to obtain a survey result that has 95 percent certainty of being within 2 percent of the nationwide preference for either one of the candidates?

p_{est} should be set at 0.5 since the election is considered even.

p_{est} = 0.5

q_{est} = 1 – p_{est} = 0.5

Half-width of the confidence interval = Margin of Error = 2 percent = 0.02

n = Roundup**(** **(** NORM.S.INV^{2} (1 – α/2) * p_{est} * q_{est} **)** / **(**Half-width of C.I.**)**^{2} **)**

n = Roundup**(** **(** NORM.S.INV^{2} (1 – 0.05/2) * 0.50 * 0.50 **)** / **(**0.02**)**^{2} **)**

n = Roundup**(** **(** NORM.S.INV^{2} ( 0.975) * 0.250 **)** / **(**0.02**)**^{2} **)**

n = Roundup**(**2400.912**)**

n = 2401

The preferences of at least 2,401 voters would have to be randomly surveyed to obtain a sample proportion that has 95 percent certainty of being within 2 percent of the national voter preference for one of the candidates.

### Example 2 of Calculating Min Sample Size in Excel

**Min Number of Production Samples to Limit Defect Rate Estimate Error Margin**

A production line is estimated to have a defect rate of approximately 15 percent of all units produced on the line. What would be the minimum number of completed production units that should be randomly sampled for defects to obtain a sample proportion of defective units that has 95 percent certainty to being within 1 percent of the real defect rate of all unites produced on that production line?

p_{est} should be set more conservatively than its estimate. The more conservative that p is, the higher will be the minimum sample size required. The most conservative setting for p_{est} would 0.5. p_{est} should be set between its estimate of 0.15 and 0.5. A reasonable setting for p_{est} would be 0.25.

p_{est} = 0.25

q_{est} = 1 – p_{est} = 0.75

Half-width of the confidence interval = Margin of Error = 1 percent = 0.01

n = Roundup**(** **(** NORM.S.INV^{2} (1 – α/2) * p_{est} * q_{est} **)** / **(**Half-width of C.I.**)**^{2} **)**

n = Roundup**(** **(** NORM.S.INV^{2} (1 – 0.05/2) * 0.25 * 0.75 **)** / **(**0.01**)**^{2} **)**

n = Roundup**(** **(** NORM.S.INV^{2} ( 0.975) * 0.1875 **)** / **(**0.01**)**^{2} **)**

n = Roundup**(**7202.735**)**

n = 7203

At least 7,203 completed units should be randomly sampled from the production line to obtain a sample proportion defective that has 95 percent certainty of being within 1 percent of the actual proportion defective of all units produced on that production line. If p_{est} were set at 0.15 instead of the more conservative 0.25, the minimum sample size would have been reduced to 4,898.

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- Overview of the 1-Sample t-Test in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- Overview of 2-Independent-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2-Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- Overview of z-Based Confidence Intervals of a Population Mean in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Simple Linear Regression Example in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Multiple Linear Regression Example in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression Performed in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer-Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA Example in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared (?2) in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi (?) – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared (?2) in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel

- Post Hoc Testing in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
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