Statistical Estimation

 

The formulas used in this chapter are the same as from the normal distribution, just re-arranged.  There are two basic types of problems, calculating a confidence interval and determining the required n for a sample.  The first set of formulas are used for calculating a confidence interval.  There are three formulas, use the first formula if the population standard deviation is available.  Use the second formula for the t distribution if the population standard deviation is not available and the sample standard deviation is used to estimate it.  The table for the t-statistic is included below.  Use the third formula when dealing with proportional data.  Small population formulas are not included here.

 

When s the population standard deviation is known use the formula

 

                 

 

 

Data from the normal table is reorganized for use in this equation as:

Reliability or

Confidence level

1 - s

Normal Deviate

za/2

.80

1.28

.90

1.64

.95

1.96

.98

2.33

.99

2.57

.998

3.08

.999

3.27

 

For example, to calculate a 95% confidence interval for a sample mean of 10 taken from a sample of n = 25 and a population standard of 5:  m = 10 ± 1.96(5/5) = 10 ± 1.96.

 

 

The table above also applies for proportional data.  Use the following formula when calculating the confidence interval.

 

 

For example, to calculate a 95% confidence interval for a sample mean of 0.4 taken from a sample of n = 25:  p = 0.4 ± 1.96(.098) = 0.4 ± 0.192

 

 

 

When the population standard deviation is not available and the sample standard deviation is used to calculate the confidence interval, the t-statistic should be used:

 

Use the t-table below, Table F in the text, to find values of t.  There are two calculations needed to use the table.  First, degrees of freedom is n - 1.   Then take the allowable error a and divide this value in two (since the table only provides one side of the two sided confidence interval).

 

For example, to calculate a 95% confidence interval for a sam1ple mean of 10 taken from a sample of n = 25 when the sample standard deviation is s = 5:  m = 10 ±  2.064(5/5) = 10 ± 2.064.  The value of the t-statistic (2.064) came by getting degrees of freedom (25 - 1) and since a 95% reliability is desired the acceptable error is .05, divide .05 in half for .025.  On the t-table below find the value in the column for a = .025   (.05/2).

 

 

Calculating required sample size.  Use the formulas below to calculate the required sample size n to achieve some target acceptable error  e:

 

 

For example, to calculate the required sample size for getting a 95% confidence interval when the population standard deviation is 5 and the tolerable error is 3:

 

n = (1.962 (52))/32  =  10.67       for 11,  always round these solutions up to the next integer

 

 

If using proportional data use:

 

 

For example, to calculate the required sample size for getting a 95% confidence interval when the population proportion is expected to be 0.5 and the tolerable error is 3%:

 

n = (1.962(.5)(1 - .5))/.032  =  1067.11    for 1068, always round up

 

 

 

 

 

This table provides the values of the t-statistic for the area in one tail of the distribution for a

 

Degrees of

Freedom

a =

.4

 

.25

 

.1

 

.05

 

.025

 

.01

 

.005

 

.0025

 

.001

 

.0005

1

.325

1.000

3.078

6.314

12.71

31.82

63.65

127.3

318.3

636.6

2

.289

.816

1.886

2.920

4.303

6.965

9.925

14.09

22.32

31.6

3

.277

.765

1.638

2.353

3.182

4.541

5.841

7.453

10.21

12.92

4

.271

.741

1.533

2.132

2.776

3.747

4.604

5.598

7.173

8.610

5

.267

.727

1.476

2.015

2.571

3.365

4.032

4.773

5.893

6.869

6

.265

.718

1.440

1.943

2.447

3.143

3.707

4.317

5.208

5.959

7

.263

.711

1.415

1.895

2.365

2.998

3.499

4.029

4.785

5.408

8

.262

.706

1.397

1.860

2.306

2.896

3.355

3.833

4.501

5.041

9

.261

.703

1.383

1.833

2.262

2.821

3.25

3.690

4.297

4.781

10

.260

.700

1.372

1.812

2.228

2.764

3.169

3.581

4.144

4.587

11

.260

.697

1.363

1.796

2.201

2.718

3.106

3.497

4.025

4.437

12

.259

.695

1.356

1.782

2.179

2.681

3.055

3.428

3.93

4.318

13

.259

.694

1.350

1.771

2.160

2.650

3.012

3.372

3.852

4.221

14

.258

.692

1.345

1.761

2.145

2.624

2.977

3.326

3.787

4.140

15

.258

.691

1.341

1.753

2.131

2.602

2.947

3.286

3.733

4.073

16

.258

.690

1.337

1.746

2.120

2.583

2.921

3.252

3.686

4.015

17

.257

.689

1.333

1.740

2.110

2.567