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. Goset (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 only for sample sizes of at most 30. With graphing calculators and computers, the practice now 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 an approximately a normal distribution with mean μ and unknown population standard deviation σ and calculate the t-score: [latex]\displaystyle{t}=\frac{{\overline{{x}}-\mu}}{{\frac{{s}}{\sqrt{{n}}}}}[/latex] 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. Because the sum of the deviations is zero, 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 zero and the distribution is symmetric about zero.
• 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 of 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 that is a completely separate assumption from normality.

Calculators and computers can easily calculate any Student’s t-probabilities.

Excel has a T.DIST function that finds probabilities for left-tailed regions of the Student’s t-distribution.  The function syntax is T.DIST(x, deg_freedom, cumulative) where [latex]x[/latex] is the numeric value at which to evaluate the distribution, deg-freedom is the degrees of freedom, and cumulative is a logical value that determines the form of the function and takes either TRUE or FALSE (or similarly 1 or 0). As with NORM.DIST in previous sections, we will only use cumulative = TRUE or 1).

The T.INV function in Excel is used to find the value of [latex]t[/latex] if we know the probability.  The syntax is T.INV(probability, deg_freedom) where probability is the area to the left of the desired [latex]t[/latex] and deg_freedom is the degrees of freedom.

The TI-83,83+, 84 and 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 specified.

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). When using a 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 (using T as the random variable) is:

• T ~ t_df 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:

• EBM = [latex]\displaystyle({t}_{\frac{{\alpha}}{{2}}})(\frac{{s}}{{\sqrt{n}}})[/latex]
• [latex]\displaystyle({t}_{\frac{{\alpha}}{{2}}})[/latex] is the t-score with area to the right equal to[latex]\displaystyle\frac{{\alpha}}{{2}}[/latex],
• use df = n – 1 degrees of freedom, and
• s = sample standard deviation.

The format for the confidence interval is:

([latex]\displaystyle\overline{x}[/latex] – EBM, [latex]\displaystyle\overline{x}[/latex]+EBM)

or

[latex]\overline{x}[/latex]– EBM  ≤ μ ≤  [latex]\overline{x}[/latex] + EBM

Calculate the Confidence Interval Directly

Using Excel:

• Use function CONFIDENCE.T to calculate the EBM.  The function has syntax CONFIDENCE.T(alpha, standard_dev, size) where alpha is the significance level (1 – CL), the standard deviation is the population standard deviation (or estimated by the sample standard deviation, s), and size is the sample size n.
• Construct confidence interval using EBM.

Using TI83/84:

• Press STAT.
• Arrow over to TESTS. Arrow down to 8:TInterval and press ENTER (or just press 8).
• Arrow toData and pressENTER.
• Arrow down and enter the name of the list where the data is stored.
• EnterFreq: 1Enter C-Level
• Arrow down toCalculateand pressEnter

Note: When calculating the error bound, a probability table for the Student’s t-distribution can also be used to find the value of t. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row); the t-score is found where the row and column intersect in the table.

References

“America’s Best Small Companies.” Forbes, 2013. Available online at http://www.forbes.com/best-small-companies/list/ (accessed July 2, 2013).

Data from Microsoft Bookshelf.

Data from http://www.businessweek.com/.

Data from http://www.forbes.com/.

“Disclosure Data Catalog: Leadership PAC and Sponsors Report, 2012.” Federal Election Commission. Available online at http://www.fec.gov/data/index.jsp (accessed July 2,2013).

“Human Toxome Project: Mapping the Pollution in People.” Environmental Working Group. Available online at http://www.ewg.org/sites/humantoxome/participants/participant-group.php?group=in+utero%2Fnewborn (accessed July 2, 2013).

“Metadata Description of Leadership PAC List.” Federal Election Commission. Available online at http://www.fec.gov/finance/disclosure/metadata/metadataLeadershipPacList.shtml (accessed July 2, 2013).

Concept Review

In many cases, the researcher does not know the population standard deviation, σ, of the measure being studied. In these cases, it is common to use the sample standard deviation, s, as an estimate of σ. The normal distribution creates accurate confidence intervals when σ is known, but it is not as accurate when s is used as an estimate. In this case, the Student’s t-distribution is much better.

Define a t-score using the following formula: [latex]\displaystyle\frac{{\overline{x}-\mu}}{{\frac{{s}}{{\sqrt{n}}}}}[/latex] 

The t-score follows the Student’s t-distribution with n – 1 degrees of freedom. The confidence interval under this distribution is calculated with EBM = [latex]\displaystyle{t}_{\frac{{\alpha}}{{2}}}(\frac{{s}}{{\sqrt{n}}})[/latex] where [latex]\displaystyle{t}_{\frac{{\alpha}}{{2}}}[/latex] is the t-score with area to the right equal to [latex]\displaystyle\frac{{\alpha}}{{2}}[/latex] s is the sample standard deviation, and n is the sample size. Use a table, calculator, or computer to find [latex]\displaystyle{t}_\frac{{\alpha}}{{2}}[/latex] for a given confidence level.

Formula Review

s = the standard deviation of sample values.

t = [latex]\displaystyle\frac{{\overline{x}-\mu}}{{\frac{{s}}{{\sqrt{n}}}}}[/latex] is the formula for the t-score which measures how far away a measure is from the population mean in the Student’s t-distribution

df = n – 1; the degrees of freedom for a Student’s t-distribution where n represents the size of the sample

T~t_df the random variable, T, has a Student’s t-distribution with df degrees of freedom

EBM:   [latex]\displaystyle{t}_{\frac{{\alpha}}{{2}}}(\frac{{s}}{{\sqrt{n}}})[/latex] = the error bound for the population mean when the population standard deviation is unknown.

The general form for a confidence interval for a single mean, population standard deviation unknown, Student’s t is given by (lower bound, upper bound)

= (point estimate – EBM, point estimate + EBM) = [latex]\displaystyle(\overline{x} -{t}_{\frac{{\alpha}}{{2}}}(\frac{{s}}{{\sqrt{n}}}),\overline{x} +{t}_{\frac{{\alpha}}{{2}}}(\frac{{s}}{{\sqrt{n}}})[/latex]