Small Sample Confidence Intervals About a Mean

Interval Estimates for Small Samples:

The Student's t-distribution

When the sample size is small, the critical values for confidence intervals are determined by the Student's t-distribution, so they are called t-values rather than z-values.

The probabilities for this distribution are defined strictly by "degrees of freedom" or
the number of data values available to estimate the population's standard deviation.

A sample of size n has n – 1 degrees of freedom.

The t-value formula is identical to the one for the z-value

...... s = sample standard deviation

Confidence Interval Estimate for µ when .

Estimation with Larger Samples and Student's t -distribution:

The Student's t -distribution is generally used on small samples with n < 30. An increase in the sample size affects both the stardard error and the number of degrees of freedom. As the degrees of freedom increase, the t-value approaches the z-value for the same level of confidence. When the sample size becomes extremely large, the t-distribution converges to the z-distribution. If n > 30 use the Normal Distribution.

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Example:

142,600 167,800 136,500 108,300 126,400 133,700 162,000 149,400

Solution: n = 8 ...... ...... s = 19, 200 (by formula for sample st. dev.)

140,800 ± 16,000 =

the confidence interval is 124,700 < µ < 156, 900.

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Formulae for Small Sample Confidence Intervals

Property Confidence Interval for Mean ( )
Sample Size

when
use n – 1 degrees of freedom

when is unknown

Error

when .

 
Parameters

use s when is unknown

 

Practice (view student's t-table)

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Solutions

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Now t = 5.841 for a 99% interval:

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Student's t Distribution Probabilities (t-scores)

Conf. Level 0.8 0.9 0.95 0.98 0.99
One Tail 0.10 0.05 0.025 0.01 0.005
Two Tails 0.20 0.10 0.05 0.02 0.01
df Values of t
1 3.078 6.314 12.71 31.82 63.66
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8  1.397 1.860 2.306 2.896  3.355
9 1.383 1.833 2.262  2.821  3.250 
10 1.372  1.812 2.228  2.764  3.169 
11  1.363  1.796 2.201  2.718  3.106 
12 1.356 1.782  2.179  2.681  3.055 
13  1.350   1.771  2.160  2.650  3.012 
14  1.345   1.761  2.145  2.624  2.977 
15  1.341   1.753  2.131  2.602  2.947 
16  1.337   1.746  2.120  2.583  2.921 
17  1.333   1.740  2.110  2.567  2.898 
18 1.330  1.734  2.101  2.552  2.878 
19 1.328  1.729  2.093  2.539  2.861 
20  1.325   1.725  2.086  2.528  2.845 
21  1.323   1.721  2.080  2.518  2.831 
22 1.321  1.717  2.074  2.508  2.819 
23 1.319  1.714  2.069  2.500  2.807 
24 1.318  1.711  2.064 2.492  2.797 
25  1.316  1.708  2.060  2.485 2.787 
26 1.315  1.706  2.056  2.479  2.779 
27   1.314   1.703  2.052  2.473  2.771 
28  1.313   1.701  2.048  2.467  2.763 
29   1.311   1.699  2.045  2.462  2.756 
30 1.310  1.697  2.042  2.457  2.750 

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