## Finding Critcal Values and Degrees of Freedom

Many students struggle to find critical values of z, t, and Chi-square for a hypothesis tests. No matter how often I show them how to use the StatCrunch calculators to do this, they gravitate back to the tables. And make a lot of mistakes. To help a bit, I have created some Excel calculators that should make it a bit easier. A minimum of data/info is needed to use the calculators. First, you need to know which distribution you need and I have covered this in other posts. For the z, you just need to know alpha and the math operator in the alternative hypothesis. For the t, things get a bit more complicated as you also need to know the sample size(s) and whether or not the variances are assumed to be equal if you are dealing with a two-sample test. I have embedded the Critical Values workbook below.

Note you can use these calculators to find the degrees of freedom for the different tests too.

## Setting up Hypothesis Tests – Three Useful Tables

Step 1: **Words to Symbols Table**

Read the problem carefully, looking for key phrases and words which will identify the claim. Look for those key phrases in the table below to determine if the claim is the null or alternative hypotheses. It can be either. Because the null and alternative must be complements, finding one will also give you the other as they will be in the same row.

For an example, one problem includes this sentence: “The test results prompts a state school administrator to declare that the mean score for the state’s 8th graders is more than 260.” The “more than” phrase is a key to recognizing the claim is “more than” which is in the Alternative Hypothesis column in Row 3. The Math Operator in that cell is >. Thus, the claim is: µ > 260. The math operator in the claim always indicates the tail of the test, e.g. a **>** symbol points to the right, so this is a right tail test.

In the center column on that row, row 3, you see the two hypotheses statements written in mathematical format. You just need to substitute the claimed value for x to complete them: **H _{0}: **

**μ**

**≤**

**260, H**

_{a}:**μ**

**> 260.**

Download this table here: Words to Math Operators table 4-4-2018

**Tail of the Test Table**

The tail of the hypothesis test is determined by looking at the math operator in the alternative hypothesis. I like to think of the direction in which the math operator is pointing. A less than operator, <, seems to be pointing to the left in my mind. Thus, it indicates a left tail test. A greater than symbol, >, points right and indicates a right tail test. The not equal, ≠, does not point either way, so that clues me to think of a two-tail test.

Use this table to help you remember this:** **

Download this table here: Tail of the Test Table

The following images show a left tail, two tail, and right tail sketches for a significance level, alpha, of 0.05.

For a left or right tail test, we put all of alpha in that tail, but in the two-tail test, we put half of alpha, or 0.025 in this example, in each tail. Using those probabilities, we find the critical values.

The critical z-value in the left tail test is -1.645 and any z test statistic that is smaller or to the left of that value falls in the rejection area. Similarly, any z test statistic greater than +1.645 falls to the right of the critical z-value and is in that rejection area.

For the two-tail test, the test statistic must have a larger absolute value to fall in one of the rejection areas. Here, the z test statistic must be less than -1.96 or greater than + 1.96.

The logic of why the test statistic must have a larger absolute value is that we set the level of significance, alpha, by deciding how confident we want to be in the decision we make. Here, we want to be 95% sure we make the right decision or just have a 5% chance of making a wrong decision, a Type 1 error of rejecting a true Null hypothesis. Because the total area/probability under the normal distribution must be 1, the area between the two tails must be 0.95 if the total chance of being wrong is alpha or 5%. That means that half of alpha must go into each tail.

Although the images above are for a z-test, similar sketches could be made for a t or a chi-square test.

**How to State the Conclusion**

After you have completed the statistical analysis and decided to reject or fail to reject the Null hypothesis, you need to state your conclusion about the claim. To get the correct wording, you need to recall which hypothesis was the claim. If the claim was the null, then your conclusion is about whether there was sufficient evidence to reject the claim. Remember, we can never prove the null to be true, but failing to reject it is the next best thing.

If the claim is the alternative hypothesis, your conclusion can be whether there was sufficient evidence to support (prove) the alternative is true.

Use the following table to help you make a good conclusion.

Download this table here: How to State the Conclusion Table

The best way to state the conclusion includes the significance level of the test and a bit about the claim itself. For example, “At the 5% significance level, there is sufficient evidence to support the claim that the mean score on the test was at least 85.”

## Minimum Sample Size for Population Mean given c, sigma, E

Enter your data in the blue cells and the minimum sample size will be in the yellow cell at the bottom. Note: the standard deviation and E must be in the same units (e.g. inches, feet, etc).

Download a copy of the workbook Min Sample Size for Mean CI V1.00

## Confidence Interval for a Mean using StatCrunch

If you are given summary data and asked to find a confidence interval, it is relatively easy using StatCrunch. Here is a typical problem:

In this problem, you are told to use the t-distribution, but that may not always be the case. If you are given the sample standard deviation, s, instead of the population standard deviation, σ , use the t-distribution to find the confidence interval. In StatCrunch, this is in the T Stats menu. If you are given σ or are told to use the z-distribution, it is essentially the same steps except you select Z Stats in step 2.

Download a PDF for your notes: Confidence Interval for a Mean using StatCrunch

## Minimum Sample Size Calculator for Population Proportion Confidence Interval

If you are conducting a survey, you will need to decide on the number of people you need to sample. That, of course, depends upon how confident you want to be and how accurately you want to predict the population proportion based on your survey results. If you want to be 95% confident, that is your level of confidence, C. If you want the population proportion within + or – 2%, that is the margin of error, E, you will accept. With that information, you can find the minimum number of people to be surveyed.

If you have an estimate of the population proportion, use that as p-hat. If you do not have any prior information, use 0.5 as your initial estimate of p-hat as that will give you the largest minimum sample size.

You can do that using StatCrunch (here), but I find it quicker to use my Excel calculator below. Just enter your data in the blue cells and the answer will be in the yellow cell.

Here is the answer to part a, with no prior estimate of p-hat. The sample size is 752.

If you use the estimate of 0.34 instead of 0.5, the minimum sample size decreases to 675. The sample size decreased, which is what you should expect. Using 0.5 as the estimate when you don’t have other information will give the largest sample size.

Download a copy of the workbook here: Minimum Sample Size for Proportion CI V1.00