CK-12 Probability and Statistics - Advanced

7.6. Student’s t-Distribution http://www.ck12.org

3. The sampling distribution is moderately skewed and the sample size is 16≤n≤30.


4. The sample size is greater than 30, without outliers.

Thet-distribution also has some unique properties. These properties are:

1. The mean of the distribution equals zero.


2. The population standard deviation is unknown.


3. The variance is equal to the degrees of freedom divided by the degrees of freedom minus 2. This means that the
   degrees of freedom must be greater than two to avoid the expression being undefined.

Variance=

df

df− 2

and df> 2

4. The variance is always greater than one, although it approaches 1 as the degrees of freedom increase. This is due
   to the fact that as the degrees of freedom increase, the distribution is becoming more of a normal distribution.


5. Although the Studentt-distribution is bell-shaped, the smaller sample sizes produce a flatter curve. The distribu-
   tion is not as mounded as a normal distribution and the tails are thicker. As the sample size increases and approaches
   30, the distribution approaches a normal distribution.


6. The population is unimodal and symmetric.

Example:

Duracell manufactures batteries that the CEO claims will last 300 hours under normal use. A researcher randomly
selected 15 batteries from the production line and tested these batteries. The tested batteries had a mean life span
of 290 hours with a standard deviation of 50 hours. If the CEO’s claim were true, what is the probability that 15
randomly selected batteries would have a life span of no more than 290 hours?

Solution:

t=

x ̄−μ

s/

√

n

The degrees of freedom are(n− 1 ) = 15 − 1. This means 14 degrees of freedom.

t=

290 − 300

50 /

√

15

t=

− 10

12. 9099

t=−. 7745993

Using the graphing calculator or a table of values, the cumulative probability is 0.286, which means that if the true
life span of a battery were 300 hours, there is a 28.6% chance that the life span of the 15 tested batteries would be
less than or equal to 290 days. This is not a high enough level of confidence to reject the null hypothesis and count
the discrepancy as significant.

You are now on theY=screen.

Y=tpdf(−. 7745993 , 14 ) = [ 0. 286 ]