Page 383 - DMTH404_STATISTICS

Unit 27: T - Distributions

Problem 2 Notes

Suppose scores on an IQ test are normally distributed, with a mean of 100. Suppose 20 people are
randomly selected and tested. The standard deviation in the sample group is 15. What is the
probability that the average test score in the sample group will be at most 110?
Solution:
To solve this problem, we will work directly with the raw data from the problem. We will not
compute the t score; the T Distribution Calculator will do that work for us. Since we will work
with the raw data, we select “Sample mean” from the Random Variable dropdown box. Then,
we enter the following data:
 The degrees of freedom are equal to 20 - 1 = 19.
 The population mean equals 100.

 The sample mean equals 110.
 The standard deviation of the sample is 15.
We enter these values into the T Distribution Calculator. The calculator displays the cumulative
probability: 0.996. Hence, there is a 99.6% chance that the sample average will be no greater than
110.

27.3 Summary

 Let X , X ...... X be n independent random variables from a normal population with mean
1 2 n
m and standard deviation s (unknown).
When s is not known, it is estimated by s, the sample standard deviation
æ 1 2 ö
ç s = å ( X - X ) ÷ . In such a case we would like to know the exact distribution of
i
-
è n 1 ø
-
X m
the statistic and the answer to this is provided by t - distribution.
s/ n
-
X m
W.S. Gosset defined t statistic as t = which follows t - distribution with (n - 1)
s/ n
degrees of freedom.
2
 Like c - distribution, t - distribution also has one parameter n = n - 1, where n denotes
sample size. Hence, this distribution is known if n is known.
n
 Mean of the random variable t is zero and standard deviation is , for n > 2.
n - 2
 The probability curve of t - distribution is symmetrical about the ordinate at t = 0. Like a
normal variable, the t variable can take any value from -  to .
 The distribution approaches normal distribution as the number of degrees of freedom
become large.
 The random variate t is defined as the ratio of a standard normal variate to the square root
2
of c - variate divided by its degrees of freedom.

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