http://www.ck12.org Chapter 7. Sampling Distributions and Estimations
7.6 Student’s t-Distribution
Learning Objectives
- Use Student’st-distribution to estimate population mean interval for smaller samples.
- Understand how the shape of Student’st-distribution corresponds to the sample size (which corresponds to a
measure called the “degrees of freedom.”)
Introduction
In a previous lesson you learned about the Central Limit Theorem. One of the attributes of this theorem was that
the sampling distribution of sample mean will follow a normal distribution as long as the sample size is large. As
the value ofnincreases, the sampling distribution is more and more likely to follow a normal distribution. You’ve
also learned that when the standard deviation of a population is known, az-score can be calculated and used with
the normal distribution to evaluate probabilities with the sample mean. In real-life situations, the standard deviation
of the entire population(σ), is rarely known. Also the sample size is not always large enough to emulate a normal
distribution. In fact there are often times when the sample sizes are quite small. What do you do when either one or
both of these events occur?
t-Statistic
People often make decisions from data by comparing the results from a sample to some hypothesized or predeter-
mined parameter. These decisions are referred to as tests of significance or hypothesis tests since they are used to
determine whether the predetermined parameter is acceptable or should be rejected. We know that if we flip a fair
coin, the probability of getting heads is 0.5. In other words, heads and tails are equally likely. Therefore, when a
coin is spun, it should land heads 50% of the time. Let’s say that a coin of questionable fairness was spun 40 times
it landed heads 12 times. For these spins the sample proportion of heads is ˆp=^1240 = 0 .3. If technology is used to
determine a 95% confidence interval to support the standard that heads should land 50% of the time, the reasonably
likely sample proportions are in the interval 0.34505 to 0.65495. The class with ˆp= 0 .3, is not captured within this
confidence interval. Therefore, the fairness of this coin should be questioned; or, in other words, value of 0.5 as a
plausible value for the proportion of times this particular coin lands heads when it is spun should be rejected. This
data has provided evidence against the standard.
The object is to test the significance of the difference between the sample and the parameter. If the difference is
small (as defined by some predetermined amount), then the parameter is acceptable. The statement that the proposed
parameter is true is called the null hypothesis. If the difference is large and can’t reasonably be attributed to chance,
then the parameter can be rejected.
When the sample size is large, reliable estimates of the mean and variance of the population from which the sample
was drawn can be made. Up to this point, we have used thez-score to determine the number of standard deviations
a given value lays above or below the mean.