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:

To test the durability of a new paint used for the white lines on the highway, the company paints 8 strips of it on a busy road and counts the number of "crossings" it takes to begin deteriorating the paint surface. Rounded to the nearest hundreth, here's the data:

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

Construct a 95% confidence interval for the average number of crossings it takes to deteriorate the paint surface. Round to the nearest hundred.

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

Since n = 8, there are 8 – 1 = 7 degrees of freedom at 95%
The critical t-value for 95% and 7 degrees of freedom = 2.365.

140,800 ± 16,000 =

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

The surface of the paint will begin to deteriorate after the lines have been crossed between 124,700 and 156,900 times.

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Formulae for Small Sample Confidence Intervals

 Property Confidence Interval for Mean ( ) Sample Size when use n – 1 degrees of freedomwhen is unknown Errorwhen . Parametersuse s when is unknown

Practice (view student's t-table)

1.

Tim weighed himself once a week for several years. Last month his four measurements (in pounds) were: ...... 190.5 ......189.0 ......195.5 ......187.0
a) Construct a 90% confidence interval for his mean weight for last month.
b) Suppose that Tim wants 99% confidence rather than 90%. Reconstruct the confidence interval for all 4 measurements.
c) Tim now wants to estimate his monthly weight accurate to within 2 pounds, with 95% confidence. What sample size does he need to achieve this?

2.

The personnel department of a large corporation wants to estimate the family dental expenses of its employees to determine the feasibility of providing a dental insurance plan.
A random sample of ten employees reveals that the family dental expenses for the preceding year had a mean of \$261 and a standard deviation of \$139.
Set up a 90% confidence interval estimate of the mean family dental expenses for all employees of this corporation.

3.

A random sample of 16 summer days in Montreal, showed the mean level of CO was 4.9 ppm. The standard deviation was 1.2 ppm. Based on this, construct a 95% confidence interval estimate of the true mean level of CO in summer in Montreal.

Solutions

1.

Tim weighed himself once a week for several years. Last month his four measurements (in pounds) were: ...... 190.5 ......189.0 ......195.5 ......187.0
a) Construct a 90% confidence interval for his mean weight for last month.
Solution: with 4 sample statistics, we'll use the t-distribution.
data:
with 3 degrees of freedom and alpha = 10%, the t-value is 2.353
the 90% confidence interval for his mean weight is (186.23, 194.77).
b) Suppose that Tim wants 99% confidence rather than 90%. Reconstruct the confidence interval for all 4 measurements.

Solution:

Now t = 5.841 for a 99% interval:

So the 99% confidence interval is (179.9, 201.1)

c) Tim now wants to estimate his monthly weight accurate to within 2 pounds, with 95% confidence. What sample size does he need to achieve this?
Solution:
Now t = 3.182 for a 95% with 3 dof :
a sample of 34 measurements is required.

2.

The personnel department of a large corporation wants to estimate the family dental expenses of its employees to determine the feasibility of providing a dental insurance plan.
A random sample of ten employees reveals that the family dental expenses for the preceding year had a mean of \$261 and a standard deviation of \$139.
Set up a 90% confidence interval estimate of the mean family dental expenses for all employees of this corporation.

Solution:
data: n = 10 ............ s = \$139 ......9 dof ...... t-value = 1.833,
= 261 ± 80.57
\$ 180.43 < µ < \$ 341.57

3.

A random sample of 16 summer days in Montreal, showed the mean level of CO was 4.9 ppm. The standard deviation was 1.2 ppm. Based on this, construct a 95% confidence interval estimate of the true mean level of CO in summer in Montreal.

Solution:
Since n = 16, this a small sample t-distribution situation.
We know that , s = 1.2, and
degrees of freedom = 15, so ta/2, 15 = 2.131,
so our confidence limits are 4.9 ± 2.131(0.3) = 4.9 ± 0.639
the confidence interval for the mean level of CO is [ 4.26, 5.54 ] parts per million.

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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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