Note about Software and Hypothesis Testing! In general, as we will learn, software usually performs tests using the p-value method. That is, the output from software will provide the test statistic and the p-value, along with some other general information (e.g. a confidence interval). To perform rejection region tests you would need to find the critical values from the tables. However, at the end of this lesson, we do demonstrate how to find the correct critical value from the standard normal distribution.
Minitab® – Conduct a One-Sample Proportion Z-Test
To conduct the one-sample proportion Z-test in Minitab...
- Click Stat > Basic Stat > 1 Proportion .
- In the drop-down box use "One or more samples, each in a column" if you have the raw data, otherwise select "Summarized data" if you only have the sample statistics.
- If using the raw data enter the column of interest into the blank variable window below the drop down selection. If using summarized data enter the number of successes for Events and the sample size for Trials .
- Click the check box for "Perform hypothesis test" and enter the null hypothesis value.
- Click Options .
- Enter the confidence level associated with alpha (e.g. 95% for alpha of 5%).
- From the drop down list for "Alternative hypothesis" select the correct alternative.
- If conditions are satisfied to perform a z-test for one proportion, select from the "Method" field "normal approximation"
- Click OK and OK .
Example 6-6: Penn State Students from Pennsylvania Section
Recall our one-proportion example at the beginning of this lesson on whether the majority of Penn State students are from Pennsylvania. In that example, we took a random sample of 500 Penn State students and found that 278 are from Pennsylvania. Can we conclude that the proportion is larger than 0.5 at a 5% level of significance? Also recall in that example we found by hand a test statistic of \(z^* = 2.504 \) and p-value of 0.0062.
Our hypotheses were: \(H_0\colon p=0.5 \) and \(H_a \colon p>0.5 \)
- Select Stat > Basic Stat > 1 Proportion.
- Choose the summarized data option and enter 278 for "Events" and 500 as the "Trials".
- Check the box for "Perform Hypothesis Test" and enter the null value of 0.5
- Click Options. With our stated alpha value of 5% we keep the default confidence level of 95.
- Select Proportion > hypothesized proportion from the Alternative Hypothesis list. Since we verified the the conditions were satisfied, select Normal Approximation under Method.
- Click OK and OK again.
The output is:
|Sample||X||N||Sample p||95% Lower Bound||Z-Value||P-Value|
As the output indicates, our by-hand calculations were very accurate!
Minitab® – Finding the Critical Value for a One-Sample Proportion Test
Although we can find values on the standard normal table, it is more accurate to find values using software. Finding values for the standard normal is discussed in more detail in Lesson 3. We present this here as a review. In order to obtain the exact critical value to use in order to conduct the rejection region approach we can use a statistical package such as Minitab.
To find the critical value...
- Choose Calc > Probability Distributions > Normal distribution
- Choose the radio button for "Inverse Cumulative Distribution" (this finds the z-value that produces the entered probability to the left of it).
- Choose the radio button for "Input constant" and enter the alpha value (if one-side alternative) or alpha/2 (if two-sided alternative).
- Click Ok