Section 4: Hypothesis Testing about a mean - small samples
In all the samples given in section 2 and 3, n > 30. If n is not greater than 30 AND you do not know the population standard deviation - you use a slightly different distribution - the t distribution. The test statistic is the same (essentially), but you compare against the t instead of the normal. Here is an example:
EXAMPLE 4: The expense of moving the storage yard for Consolidated Delivery Service is justified only if it can be shown that the mean travel distance is less than 214 miles. In trial runs of 12 delivery trucks, the mean and standard deviation are found to be 198 miles and 42 miles respectively. At the 0.01 level of significance, test the claim that the mean is less than 214 miles
Solution: Claim: 
The test statistic to use if sample size is less than 30 and you do not know the population standard deviation is the t statistic
You will notice that this statistic is essentially the same as the z statistic, what is different is that you compare against the t distribution and not the normal. So since there are 11 degrees of freedom, you distribution looks as follows - use the 0.01 ONE TAIL column to find the critical value
You are NOT in the critical region, so you should Fail to Reject the null hypothesis. Here is the conclusion:
There is not sufficient evidence to support the claim that mean distance is less than 214 miles
P-VALUE METHOD:
A P-Value or probability value is the probability of getting a value of the sample test statistic that is at least as extreme as the one found from the sample data, assuming the null hypothesis is true.
If you are not allowed to use technology (i.e. you must use a table), I suggest you use the classical method of hypothesis testing. However if you use the TI-83, it uses the p-value method
In testing using the p-value you follow the same steps, but after you calculate the test statistic, you find a p-value. You may find a p-value from the table, but it is very inaccurate. So I only use this method when using the calculator. Once you have a p-value - use this guideline:
Reject the null hypothesis if the p-value is less than or equal to the significance level. Fail to reject if p-value is greater than the significance level.
Example 2: Because of the expense involved, car crash tests often involve small samples. When 5 BMW cars are crashed under standard conditions the repair costs are shown in the accompanying table. Use a 0.05 significance level to test the claim that the mean for all BMW cars is less than $1000
| $797 | $571 | $904 | $1147 | $418 |
Since the p value is not less than the significance level - we fail to reject the null hypothesis. The final conclusion is there is not sufficient sample evidence to support the claim that the average cost is less than $1000