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
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.
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?
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.
data: n = 10 ............ s = $139 ......9 dof ...... t-value = 1.833,
= 261 ± 80.57
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.
Since n = 16, this a small sample t-distribution situation.
We know that , s = 1.2, and
degrees of freedom = 15, so t_{a/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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Statistics MathRoom Index
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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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Statistics MathRoom Index
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