In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gossett (1876-1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very few samples. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to "discover" what is called the Student's-t distribution. The name comes from the fact that Gosset wrote under the pen name "Student."
Up until the mid 1970s, some statisticians used the normal distribution approximation for large sample sizes and only used the Student's-t distribution for sample sizes of at most 30. With the common use of graphing calculators and computers, the practice is to use the Student's-t distribution whenever s is used as an estimate for σ.
If you draw a simple random sample of size n from a population that has approximately a normal distribution with mean µ and unknown population standard deviation σ and calculate the t-score , then the t-scores follow a Student's-t distribution with n − 1 degrees of freedom. The t-score has the same interpretation as the z-score. It measures how far is from its mean µ. For each sample size n, there is a different Student's-t distribution.
The degrees of freedom, n − 1, come from the calculation of the sample standard deviation s. In Chapter 2, we used n deviations ( values) to calculate s. Because the sum of the deviations is 0, we can find the last deviation once we know the other n − 1 deviations. The other n − 1 deviations can change or vary freely. We call the number n − 1 the degrees of freedom (df).
Properties of the Student's-t Distribution
- The graph for the Student's-t distribution is similar to the Standard Normal curve.
- The mean for the Student's-t distribution is 0 and the distribution is symmetric about 0.
- The Student's-t distribution has more probability in its tails than the Standard Normal distribution because the spread of the t distribution is greater than the spread of the Standard Normal. So the graph of the Student's-t distribution will be thicker in the tails and shorter in the center than the graph of the Standard Normal distribution.
- The exact shape of the Student's-t distribution depends on the "degrees of freedom". As the degrees of freedom increases, the graph Student's-t distribution becomes more like the graph of the Standard Normal distribution.
- The underlying population of individual observations is assumed to be normally distributed with unknown population mean µ and unknown population standard deviation σ. The size of the underlying population is generally not relevant unless it is very small. If it is bell shaped (normal) then the assumption is met and doesn't need discussion. Random sampling is assumed but it is a completely separate assumption from normality.
Calculators and computers can easily calculate any Student's-t probabilities. The TI-83,83+,84+ have a tcdf function to find the probability for given values of t. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.
For the TI-84+ you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm. The invT command requires two inputs: invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specifed.
The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)
A probability table for the Student's-t distribution can also be used. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's-t distribution.) When using t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.
A Student's-t table gives t-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's-t probabilities.
The notation for the Student's-t distribution is (using T as the random variable) is
- T ∼ tdf where df = n − 1.
- For example, if we have a sample of size n = 20 items, then we calculate the degrees of freedom as df = n−1 = 20−1 = 19 and we write the distribution as T ∼ t19
If the population standard deviation is not known, the error bound for a population mean is:
- is the t-score with area to the right equal to
- use df = n − 1 degrees of freedom
- s = sample standard deviation
The format for the confidence interval is:
The TI-83, 83+ and 84 calculators have a function that calculates the confidence interval directly. To get to it,
Arrow over to TESTS.
Arrow down to 8:TInterval and press ENTER (or just press 8).
Suppose you do a study of acupuncture to determine how Effective it is in relieving pain. You measure sensory rates for 15 subjects with the results given below. Use the sample data to
construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) from which you took the data.
The solution is shown step-by-step and by using the TI-83, 83+ and 84+ calculators. 8.6; 9.4; 7.9; 6.8; 8.3; 7.3; 9.2; 9.6; 8.7; 11.4; 10.3; 5.4; 8.1; 5.5; 6.9
- You can use technology to directly calculate the confidence interval.
- The first solution is step-by-step (Solution A).
- The second solution uses the Ti-83+ and Ti-84 calculators (Solution B).
To find the confidence interval, you need the sample mean, , and the EBM.
= 8.2267 s = 1.6722 n = 15
df = 15 − 1 = 14
CL = 0.95 so α = 1 − CL = 1 − 0.95 = 0.05
The area to the right of t.025 is 0.025 and the area to the left of t.025 is 1−0.025 = 0.975
using invT(.975,14) on the TI-84+ calculator.
The 95% confidence interval is (7.30, 9.15).
We estimate with 95% confidence that the true population mean sensory rate is between 7.30 and 9.15.
Using a function of the TI-83, TI-83+ or TI-84 calculators:
Press STAT and arrow over to TESTS.
Arrow down to 8:TInterval and press ENTER (or you can just press 8).
Arrow to Data and press ENTER.
Arrow down to List and enter the list name where you put the data.
Arrow down to Freq and enter 1.
Arrow down to C-level and enter .95
Arrow down to Calculate and press ENTER.
The 95% confidence interval is (7.3006, 9.1527)
**With contributions from Roberta Bloom