Definition :
Definition :
When some samples are drawn from normal population whose variance is known, a distribution of the sample mean is normal. When, however, the variance of the population is unknown, the distribution is not normal but student-t, whose tail longer. That means the fact that sample mean with unknown population variance is inclined to be an extreme value. If you use normal distribution for hypothesis testing instead of t distribution, probability of error becomes bigger.
Formula :
Suppose we have a simple random sample of size n drawn from a Normal population with mean
and standard deviation
. Let
denote the sample mean and s, the sample standard deviation. Then the quantity
has a t distribution with n-1 degrees of freedom.
Note that there is a different t distribution for each sample size, in other words, it is a class of distributions. When we speak of a specific t distribution, we have to specify the degrees of freedom. The degrees of freedom for this t statistics comes from the sample standard deviation s in the denominator of equation 1.
The t density curves are symmetric and bell-shaped like the normal distribution and have their peak at 0. However, the spread is more than that of the standard normal distribution. This is due to the fact that in formula (1), the denominator is s rather than σ. Since s is a random quantity varying with various samples, the variability in t is more, resulting in a larger spread.
The larger the degrees of freedom, the closer the t-density is to the normal density. This reflects the fact that the standard deviation s approaches
Properties :
The Student t distribution is different for different sample sizes.The Student t distribution is generally bell-shaped, but with smaller sample sizes shows increased variability (flatter). In other words, the distribution is less peaked than a normal distribution and with thicker tails. As the sample size increases, the distribution approaches a normal distribution. For n > 30, the differences are negligible.
The mean is zero (much like the standard normal distribution).The distribution is symmetrical about the mean.
The variance is greater than one, but approaches one from above as the sample size increases (=1 for the standard normal distribution).
The population standard deviation is unknown.
The population is essentially normal (unimodal and basically symmetric)
Method :
Illustrations :
The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. A sample of 10 randomly selected hours from the last month revealed the mean hourly production on the new machine was 256, with a sample standard deviation of 6 per hour. At the .05 significance level can Neary conclude that the new machine is faster ?
Step 1: H0: M <= 250 H1: M>250
Step 2: H0 is rejected if t>1.833, df = 9
Step 3: t = [256 - 250]/[6/sqrt(10] = 3.16
Step 4: H0 is rejected. The new machine is faster.
Applications :
It is often the case that one wants to calculate the size of sample needed to obtain a certain level of confidence in survey results. Unfortunately, this calculation requires prior knowledge of the population standard deviation (
Applet :
For an illustration of T distributions go to
http://www.econtools.com/jevons/java/Graphics2D/tDist.html
Excel Function :
These functions can be accessed by clicking on Insert and then choosing Function from the drop down menu.
The Excel function for finding a Studen'ts T distribution for a given data set is :
TTEST(array1,array2,tails,type)
Returns the probability associated with a student's T test
