In our example, you would be unable to conclude that the label for the protein bars should be changed. You fail to reject the null hypothesis that the mean is equal to the specified value. The test statistic is less extreme than the critical t values in other words, the test statistic is not less than -2.042, or is not greater than +2.042.There are two possible results from our comparison: The t value for a two-sided test with α = 0.05 and 30 degrees of freedom is +/- 2.042. Statisticians write the t value with α = 0.05 and 30 degrees of freedom as: The degrees of freedom ( df) are based on the sample size and are calculated as: Using the energy bar data as an example, we set α = 0.05. We compare the test statistic to a t value with our chosen alpha value and the degrees of freedom for our data. The test statistic uses the formula shown below: The formula shows the sample standard deviation as s and the sample size as n. We calculate the average for the sample and then calculate the difference with the population mean, mu: If we cannot reject the null hypothesis, then we make a practical conclusion that the labels for the bars may be correct. If we can reject the null hypothesis that the mean is equal to 20 grams, then we make a practical conclusion that the labels for the bars are incorrect. We are testing if the population mean is different from 20 grams in either direction. The labels claiming 20 grams of protein would be incorrect. The alternative hypothesis is that the underlying population mean is not equal to 20. Our null hypothesis is that the underlying population mean is equal to 20. Let’s look at the energy bar data and the 1-sample t-test using statistical terms. We make a practical conclusion that the labels are incorrect, and the population mean grams of protein is greater than 20. Since 3.07 > 2.043, we reject the null hypothesis that the mean grams of protein is equal to 20. We compare the value of our statistic (3.07) to the t value. The most likely situation is that you will use software and will not use printed tables. Most statistics books have look-up tables for the distribution. The critical value of t with α = 0.05 and 30 degrees of freedom is +/- 2.043. For the energy bar data:ĭegrees of freedom = $ n - 1 = 31 - 1 = 30 $ The degrees of freedom are based on the sample size. For a t-test, we need the degrees of freedom to find this value. We find the value from the t-distribution based on our decision. In practice, setting your risk level (α) should be made before collecting the data. For the energy bar data, we decide that we are willing to take a 5% risk of saying that the unknown population mean is different from 20 when in fact it is not.
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