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6.7.2. Two Samples

Sample Size and Power-Two Samples

The first procedure provided here concerns estimation of the difference between two population means and as such it is different from the last three. The latter are derived from the same equation, each time solving for a different parameter.

6.7.2.1. Sample Size in Estimating Two Population Means

The sample size, which is assumed to be the same for both samples, is computed from:

   Sample Size and Power-Two Samples

Here:

·      Sample Size and Power-Two Samples is the pooled sample variance with ν degrees of freedom,

·      d is the half-width of the desired confidence interval,

·      1 - α is the confidence level,

·      1 - β is the assurance that the confidence interval will not be greater,

·      Sample Size and Power-Two Samplesis the two-tailed critical value for α from t-distribution with 2(n ‑ 1) degrees of freedom,

If the two sample sizes are not equal, and one of them is given, then the other sample size is estimated according to the following formula, which is derived from their harmonic mean:

   Sample Size and Power-Two Samples

Since critical values of t and F on the right hand side of this equation depend on the sample size n, the left hand side cannot be computed directly. Instead, an iterational algorithm is employed and usually the convergence is achieved within a few iterations.

The user is expected to enter:

·      Denominator Degrees of Freedom

·      Pooled Variance

·      Half-Width of Confidence Interval

·      Assurance for Confidence Interval

·      Confidence Level

·      n1 is given when n1 Sample Size and Power-Two Samples n2 (optional)

and the program will output the estimated sample size.

In case when the two samples are assumed to be of equal size, the value of the n1 field should be zero.

Example

Example 8.3 on p. 147 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Two Samples and the Sample Size in Estimating Two Population Means option. Enter the following data at the next dialogue:

Sample Size and Power Estimation: Two Samples

Sample Size in Estimating Two Population Means

Denominator Degrees of Freedom =

 11.0000

Pooled Variance =

 0.5193

Half-Width of Confidence Interval =

 0.5000

Assurance for Confidence Interval =

 0.9000

Confidence Level =

 0.9500

N1 Given (Optional) =

 0.0000

Estimated Sample Size =

 17.2934

 

6.7.2.2. Sample Size in Tests Concerning Two Means

The purpose of the experiment is to test if the two samples are taken from one population “H0: μ1 = μ2” or from two distinct populations “H0: μ1 Sample Size and Power-Two Samples μ2”.

The sample size is computed from:

   Sample Size and Power-Two Samples

Here:

·      Sample Size and Power-Two Samples is the pooled sample variance with ν degrees of freedom,

·      δ is the minimum detectable difference between the two population means at the desired confidence interval,

·      α is the probability of committing a Type I error and 1- α is the confidence level, which can be one or two-tailed.

·      β is the probability of committing a Type II error and 1 - β is the power of the test, which is the probability of detecting a real difference,

·      Sample Size and Power-Two Samples is the (one or) two-tailed critical value for α from t-distribution with ν degrees of freedom,

·      Sample Size and Power-Two Samples is the one-tailed critical value for β from t-distribution with ν degrees of freedom.

Since critical values on the right hand side of this equation are dependent on the sample size n, an iterational algorithm is employed.

The user is expected to enter:

·      Pooled Sample Variance

·      Minimum detectable difference

·      Power of the test

·      Confidence Level

·      1 or 2 tailed test

·      n1 is given when n1 Sample Size and Power-Two Samples n2 (optional)

and the program will output the estimated sample size.

If the two samples are assumed to have the same size, enter 0 into the N1 Given (optional) field. If the size of one of the samples is fixed, then enter the value into this field. In this case the program will calculate the second sample size from the following formula, which is derived from their harmonic mean:

   Sample Size and Power-Two Samples

Example

Example 8.4 on p. 149 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Two Samples and the Sample Size in Tests Concerning Two Means option. Enter the following data at the next dialogue:

Sample Size and Power Estimation: Two Samples

Sample Size in Tests Concerning Two Means

Pooled Variance =

 0.5200

Minimum Detectable Difference =

 0.5000

Power of the Test =

 0.9000

Confidence Level =

 0.9500

1 or 2 Tailed Test =

 2.0000

N1 Given (optional) =

 30.0000

Estimated Sample Size =

 44.7241

N2 when N1 Given =

 87.8328

 

6.7.2.3. Minimum Detectable Difference for Two Samples

Rearranging the equation in section 6.7.1.2. Sample Size in Tests Concerning the Mean we obtain:

   Sample Size and Power-Two Samples

Here, δ can be computed directly, without an iterational algorithm, as all parameters on the right hand side can be computed using the given parameters.

The user is expected to enter:

·      Sample Size

·      Pooled Sample Variance

·      Power of the test

·      Confidence Level

·      1 or 2 tailed test

and the program will output the minimum detectable difference for this sample size. For different sample sizes enter their harmonic mean:

   Sample Size and Power-Two Samples

Example

Example 8.5 on p. 150 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Two Samples and the Minimum Detectable Difference option. Enter the following data at the next dialogue:

Sample Size and Power Estimation: Two Samples

Minimum Detectable Difference

Sample Size =

 20.0000

Pooled Variance =

 0.5193

Power of the Test =

 0.9000

Confidence Level =

 0.9500

1 or 2 Tailed Test =

 2.0000

Minimum Detectable Difference =

 0.7585

 

6.7.2.4. Power of the Test for Two Samples

Rearranging the equation in section 6.7.1.2. Sample Size in Tests Concerning the Mean we obtain:

   Sample Size and Power-Two Samples

Here, Sample Size and Power-Two Samples can be computed directly and then β is obtained from the t-distribution with ν degrees of freedom.

An alternative method of estimating power of the test is also provided which is based on the noncentral F-distribution. The phi statistic is calculated from:

   Sample Size and Power-Two Samples

and its p-value is automatically calculated by the program (Pearson and Hartley (1951), pp. 112-130).

The user is expected to enter:

·      Sample Size

·      Pooled Sample Variance

·      Minimum Detectable Difference

·      Confidence Level

·      1 or 2 tailed test

and the program will output the computed t-statistic and its p-value, as well as the phi statistic and its and its p-value.

If the two samples have different sizes then their harmonic mean should be entered:

   Sample Size and Power-Two Samples

Example

Example 8.6 on p. 151 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Two Samples and the Power of the Test option. Enter the following data at the next dialogue:

Sample Size and Power Estimation: Two Samples

Power of the Test

Sample Size =

 15.0000

Pooled Variance =

 0.5193

Minimum Detectable Difference =

 1.0000

Confidence Level =

 0.9500

1 or 2 Tailed Test =

 2.0000

Power of the Test:

 

t-Statistic =

 1.7519

2-Tail Probability =

 0.9546

Phi =

 2.6872

Probability =

 0.9561

 


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