~For most beginning students, hypothesis testing is somewhat confusing. Many can go through the motions and press the right buttons on the TI-83 and get conclusions with very little understanding.

~This example (for means with a known population standard deviation), is presented in a way that could help you see how everything is related and emphasizes the understanding aspect of the problem.

~It is totally hypothetical and does not reflect true conditions, necessarily.  However, it serves my purpose.

~First of all, here are some basics:

1)  hypothesis testing involves making a decision on whether or not the "null hypothesis" (which involves a SINGLE value of the parameter) is to be rejected or not. (the population parameter tested EQUALS a certain value).

2)  when rejecting the "null hypothesis", we conclude that the sample data provided sufficient evidence (based on the location of the test statistic & significance level) to warrant this action.

3)  if we do not reject the "null hypothesis", we conclude that the sample data did not provide us with sufficient evidence to reject. (based on the level of significance used).

4)  when rejecting the "null hypothesis", we are accepting the "alternate hypothesis". (based on the level of significance used).

5)  for a given problem, an original claim is usually made.  This claim could be the "null hypothesis" or related to the "alternate hypothesis" (in most cases when it is not the "null hypothesis" it is the "alternate hypothesis")
(see my link on "hypothesis testing" for details).

6)  so, it's important to formulate the "alternate hypothesis" correctly or you will be in error. See    Choosing the alternate hypothesis

7)  the wording of your conclusion should address the original claim in some way.  A common error students make is being too vague in this respect.  Use your common sense (not easy sometimes) and be as clear as possible.

8)  remember, your decision to reject or not reject the "null hypothesis" is based on whether or not the p-value is less than the significance level used.

9)  once the significance level is chosen, our sample gives us a test statistic
(z or t value) and it's location is determined (relative to the critical value for this level), and the p-value is calculated.  Then, our decision is made.

~Here is my example:

~It has been found that the average age of current passenger vehicles, actively on the road, is 5.8 years with a standard deviation of 3.7 years (from past records).

~In a random sample of 77 such vehicles, the average age was found to be 4.7 years.

~Question:  Does this sample provide us with sufficient evidence to conclude that the mean of 5.8 years is too high? (at a given level of significance).

~for this example, the "null hypothesis" (H₀) is m₀=5.8 and the "alternate hypothesis" (H₁) is m<5.8 (our sample data suggests that).

~based on the "alternate hypothesis", this becomes a left tailed-test.

~at the .01 level of significance, what do we conclude?

~since the population standard deviation is known and the sample size is greater than 30, we will use the ZTest (menu 1 under STAT, Tests).

~go to that menu and click enter.

1)  Press stats (since the mean & standard deviation of the population is given).

2)  enter the "null hypothesis", m₀: 5.8

3)  enter the population standard deviation, s: 3.7

4)  enter the sample mean: 4.7

5)  enter n: 77

6)  enter the "alternate hypothesis", m: < m₀

7)  go to calculate, enter

~the display will show the following:

1) m < 5.8 (alternate hypothesis)

2) z = -2.60877 (test statistic from our sample mean of 4.7)

3) p = .00454 (p-value) (area to the left of our test statistic)

4) x bar = 4.7 (sample mean)

5) n = 77 (our sample size)

~all we have to do is to compare the p-value to our significance level to make our decision whether to reject or not reject the "null hypothesis".

~since the p-value is less than our significance level of .01, the test statistic (z= -2.60877) falls to the left of our critical value of z = -2.32635 determined by our significance level of .01.

~the critical value need not be calculated. The TI-83 does this for us automatically.

~if all you wanted was the critical value, just go to 2nd, Vars, menu 3, enter .01 into invnorm. For a critical t-value, use the table with the correct degee of freedom (n-1) & look under the headings for the amount of area in the the tails (determined by the significance level).

~so, since the test statistic falls within the rejection region, we will reject the "null hypothesis".

~the following is the best way to state your conclusion.  Other ways might not be acceptable to some teachers.

~Conclusion (best way):  there is sufficient evidence to conclude that the mean age of passenger vehicles on the road is less than 5.8 years.

~Another way:  there is insufficient evidence that the mean age of passenger vehicles is 5.8 years. (too vague) (more information can be given concerning the direction).

~Worst way:  we reject the "null hypothesis".  This would be unacceptable in most classrooms since it gives us no information, what so ever, about what is being tested. Always tie in the original claim in your conclusion and give as much information as you can to the reader.

~All hypothesis testing involving proportions & means are done in much the same matter. See the link "What Test to Use" under "Topics of Interest".