HYPOTHESIS TESTING

~The null hypothesis always contains the EQUAL sign.

~The alternate hypothesis is an opposing view.

~The original claim could be the null hypothesis or the alternate hypothesis.

~If the original claim states that the mean or proportion is EQUAL (=) to some value, this becomes the null hypothesis. The alternate hypothesis would take one of 3 forms,  >, <, or ≠ depending on the wording of the problem and/or common sense.

~If the original claim states that the mean or proportion is AT LEAST a value, then the alternate hypothesis would be that the original claim is LESS THAN (<) that value.    (THIS IS A LEFT-TAILED TEST)

~If the original claim states that the mean or proportion is GREATER THAN (>) a value, then this becomes the alternate hypothesis.  (THIS IS RIGHT-TAILED TEST).

~If the original claim states that mean or proportion is AT MOST a value, then the alternate hypothesis would be that the original claim is GREATER THAN (>) that value.  (THIS IS A RIGHT TAILED TEST)

~If the original claim states that the mean or proportion is LESS THAN (<) a value, then this becomes the alternate hypothesis. (THIS IS A LEFT-TAILED TEST)

~If the original claim states that the mean or proportion DIFFERS from some value, then use ≠ (TWO-TAILED TEST) for the alternate hypothesis.

~If the researcher believes that the sample suggests <, use this in the alternate hypothesis.

~If the researcher believes that the sample suggests >, use this in the alternate hypothesis.

~If there is no indication of direction, use ≠ in the alternate hypothesis.

~Critical values: These are based on the significance level you are using and whether you are using the standard normal curve or a t-curve.

~They will be z values for the standard normal curve.

~They will be t values for any t-curve you use. (there are an infinite # of t-curves)

~A value of n determines a t-curve. n-1 is the degree of freedom for any one.

~Use the standard normal curve if the population standard deviation is known.

~Use a t-curve if the population standard deviation is not known.

~These critical values (z or t values) are fixed for a given curve at a given significance level.They are located at the cut off points on the z or t axis of the rejection or shaded regions at the boundary of the tails.

~For the standard normal curve with a .05 level, the critical value for a right-tailed test (all of the .05 area is in the right tail) is z=1.645. By symmetry, for a left tailed test (all of the .05 is in the left tail), z= -1.645.

~You can find these using your TI-83 by using 2nd VARS, invNorm(.95) for the right, invNorm(.05) for the left one. InvNorm gives a z value to the right of the total area under the curve you input.

~For the standard normal curve with a .01 level, z=2.326 (for a right-tailed test), z= -2.326 for a left-tailed test. For z=2.326, that means that 99% of the area is to the left of this value & .01 or 1% of the area is to the right of this value.

~For a two-tailed test, the significance level of .05 is split between the 2 tails, so, .025 in the right-tail & .025 in the left tail.

~To find these, use InvNorm(.025) to get z= - 1.96, so the critical value at the right would be z= 1.96 (symmetry).

~If the test statistic, from our sample, falls to the right (for a right-tailed test) or left (for a left-tailed test) of the critical value, it would be in the rejection region, so we would reject the null hypothesis. If not, we do not reject the null hypothesis.

~t-curves: There are many t-curves (one for each degree of freedom, n-1)

~Your sample size n determines which one you will be using.

~For t-curves, everything above (using the standard normal curve) is exactly the same, however, the t-curves are "flatter" (less kurtosis) & their standard deviations vary from curve to curve, so, when you establish .05 or .01 areas under them (at the tails), the values at these "cutoff" will be different from curve to curve.

~These critical values can be found using table A-3. To do this, pick your significance level, & the correct degree of freedom (n-1) to determine a t-curve. Then read the critical values from the table.

~For example, with n=20, the degree of freedom is 19, at the .05 level of significance, for a right-tailed test (all of the .05 area is in the right tail), the critical value is t=1.729. So, if the test statistic is greater than 1.729, it will lie in the rejection region & we reject the null hypothesis, if not, we do not reject it, just like before.

~Remember, we only use t-curves when the standard deviation of the population is not known, otherwise, we would use the standard normal curve with z values as critical values.

~test statistic: This comes from our sample. The TI-83 will give it to you automatically. If you are using the standard normal curve, use STAT, TESTS, Z-test(1), fill in the information, Calculate.

~The display will give you the test statistic and the p-value underneath it.

~If you are using t-curves (standard deviation of the population is unknown), use STAT, TESTS, T-Test, fill in the information, Calculate. The display will give you the test statistic and its p-value underneath it.

~Now, for the p-value: It has to do with the area under the curve you are using. It gives the area to the right of the test statistic (for a right-tailed test), & the area to the left of the test statistic (for a left-tailed test).

~For a two-tailed test, it will twice the area to the right or left of the test statistic (depending upon whether or not the test statistic is positive or negative).

~In any case, IF THE P-VALUE IS LESS THAN THE SIGNIFICANCE LEVEL, WE REJECT THE NULL HYPOTHESIS, OTHERWISE, WE DO NOT REJECT IT.

~When the p-value is less than the significance level, the test statistic will fall in the rejection region. Occasionally, you'll have to compute a test statistic & p-value without the use of the Z-Test or T-test on your TI-83.

~To do this, use the formula in the insert for the test statistic, then find the p-value (area related to it--see insert) by the use of the TI-83 (2nd VARS, normalcdf (input two z values) if dealing with the standard normal curve or use table A-3, if dealing with t-curves. However, you need to read table A-3 in reverse & probably would not find your test statistic exactly in the heart of the table.

~If you find the exact value of the test statistic in the heart of the t-table, just read the area in the tail(s) for the p-value (top of chart). If not, find values on either side & get a range for the areas in the tail(s). This will give you the range in the p-value & would be sufficient, in most cases, to make your decision. If the test statistic is off the chart (too large), the p-value will be less than the smallest area at the top. If the test statistic is off the chart (too small), it will be greater than the largest area at the top. That way, you will still have a good idea of the value so you can compare it to the significance level.

~So, we often get a range of values where that p-value is located. In most cases, that would be enough to determine whether or not the p-value is less than the significance level (for rejecting or not rejecting the null hypothesis).

~In conclusion, the TI-83 will do all that is necessary to complete a hypothesis test. It doesn't give us critical t-values, but we don't need to find them since all we have to do is compare the p-value to the total area in the rejection region, which is the significance level (usually .05 or .01).

Hypothesis testing       Vs      A Courtroom trial