![]() The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row). (The TI-89 has an inverse T command.)Ī probability table for the Student's t-distribution can also be used. The TI-83 and 83+ do not have the invT command. 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 invT command works similarly to the invnorm. However for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.įor the TI-84+ you can use the invT command on the DISTRibution menu. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). The TI-83,83+, and 84+ have a tcdf function to find the probability for given values of t. Random sampling is assumed, but that is a completely separate assumption from normality.Ĭalculators and computers can easily calculate any Student's t-probabilities. If it is bell shaped (normal) then the assumption is met and doesn't need discussion. The size of the underlying population is generally not relevant unless it is very small. The underlying population of individual observations is assumed to be normally distributed with unknown population mean μ and unknown population standard deviation σ.As the degrees of freedom increases, the graph of Student's t-distribution becomes more like the graph of the standard normal distribution. The exact shape of the Student's t-distribution depends on the degrees of freedom.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 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.The mean for the Student's t-distribution is zero and the distribution is symmetric about zero.The graph for the Student's t-distribution is similar to the standard normal curve.Properties of the Student's t-Distribution We call the number n – 1 the degrees of freedom (df). The other n – 1 deviations can change or vary freely. Because the sum of the deviations is zero, we can find the last deviation once we know the other n – 1 deviations. This calculation requires n deviations ( x – x ¯ values ) ( x – x ¯ values ). The degrees of freedom, n – 1, come from the calculation of the sample standard deviation s. For each sample size n, there is a different Student's t-distribution. It measures how far x ¯ x ¯ is from its mean μ. The t-score has the same interpretation as the z-score. If you draw a simple random sample of size n from a population that has an approximately normal distribution with mean μ and unknown population standard deviation σ and calculate the t-score t = x ¯ – μ ( s n ) x ¯ – μ ( s n ), then the t-scores follow a Student's t-distribution with n – 1 degrees of freedom. ![]() 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. Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and used the Student's t-distribution only for sample sizes of at most 30. The name comes from the fact that Gosset wrote under the pen name "Student." This problem led him to "discover" what is called the Student's t-distribution. He realized that he could not use a normal distribution for the calculation he found that the actual distribution depends on the sample size. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. His experiments with hops and barley produced very few samples. Gosset (1876–1937) of the Guinness brewery in Dublin, Ireland ran into this problem. A small sample size caused inaccuracies in the confidence interval. However, statisticians ran into problems when the sample size was small. 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. ![]() In the past, when the sample size was large, this did not present a problem to statisticians. In practice, we rarely know the population standard deviation.
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