The T-Distribution is a distribution we can use when performing inference tests (confidence intervals or hypothesis tests) on the mean. There are several assumptions that need to be met before we use the T-distribution, which are listed below:

1. The population standard deviation, is unknown
2. The data comes from a distribution that is approximately Normal (This is important if we have a smaller sample size)
3. The data set contains no outliers (This is also important, as it may indicate the data is not Normal, and have a large impact on the estimate of standard deviation)

The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom (df). When you use a t-distribution to estimate the population mean, the degrees of freedom are equal to n-1. So, the critical T values used will be dependent on the sample size, and account for the additional variability from using the sample standard deviation as opposed to the population standard deviation.

When we run a hypothesis test, the over-arching process will remain the same regardless of distribution that we use. The steps to run a hypothesis test for the sample mean using the T-distribution are as follows:

Step 1: Write out your hypotheses based on the words given in the problem

Step 2: Determine your value of alpha

Step 3: Compute your T statistic

Step 4: Compare the T-statistic with the critical T-value, or compute the P-value

Step 5: Make your conclusion

If we are in the position where we do not know the population standard deviation, we want to use the T distribution. The T-statistic is found as follows: =

Now, let's look at an example. Suppose the local bakery claims their "Ginormous" cinnamon roll weighs at least 8 oz. We want to test to see whether or not this claim is true, so, we collected a sample of 15 "Ginormous" cinnamon rolls and find the average weight was 7.6 oz with a standard deviation of 1.0 ounces. Do we have enough evidence to dispute this claim? Assume the cinnamon roll weight comes from a normal distribution.

Step 1: 0:=8 vs 1:<8

Step 2: = 0.05

Step 3: Since we do not know the population standard deviation, and since we are assuming the data comes from a normal population, we can use T:

==7.681.015=1.549

Step 4: Critical T-value = -1.7613, P-value = 0.07182

Step 5: Using P-value): Our p-value is larger than our alpha value, so we will fail to reject the null hypothesis and do not have statistical evidence to dispute the claim.

(Using Critical Regions) Since our T-statistics does not fall in the critical region (T < -1.7613) we fail to reject the null hypothesis and do not have statistical evidence to dispute the claim.

Just as we did for the Z distribution, we can use the T-distribution to compute confidence intervals. If we are in a situation where the T-distribution is our distribution of choice (same assumptions previously stated), we can compute our confidence intervals using the following margin of error, E, computed as:

=

When it comes to finding critical T-values, we also need to account for the degrees of freedom. The degrees of freedom for our sample will be equal to one less than the sample size, or df = n - 1. Here is the process we follow to find critical t-values: Take the confidence level as a decimal and subtract it from 1. Then divide the resulting value by 2. Use this as our "alpha" value on our T-table. We then look up the critical T-value on our T-table for the associated degrees of freedom.

For example, if we wanted to construct a 95% confidence interval for the mean with a sample size n of 20, we would find the critical T-value as follows: 1 - 0.95 = 0.05. 0.05/2 = 0.025. This is our alpha value. Then we compute the degrees of freedom: df = 20 - 1 = 19. Now we go to our T-tables and find the value that corresponds to 19 degrees of freedom and an alpha value of 0.025, which will be 2.093

Now let's walk through a confidence interval calculation using the T distribution: Suppose we have a sample of data with a mean of 25, a sample standard deviation of 6, and a sample size of 16. We want to create a 95% confidence interval for this sample: ==2.131616=3.1965

Lower bound: =253.1965=21.8035

Upper bound: +=25+3.1965=28.1965

Then we write our final answer as such: (21.8035, 28.1965). We can then say that we are 95% confident the true value of the population mean falls between 21.8035 and 28.1965

Instructions

For this discussion post, we are going to run a hypothesis test based on a claim made by a nutrition company. Read the following:

A new weight loss medication claims that the average person taking their medication will lose at least 10 pounds in 60 days. We created an experiment where we used 20 people who took the medication and weighed them up front, then weighed them again after 60 days. The net loss is computed by taking initial weight - weight after 60 days. The following represent the individuals weight loss:

Person	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Net Loss	-2	2	18	7	13	-1	18	5	14	0	4	4	12	3	13	-1	-1	14	11	-1

Discussion Prompts

Answer the following questions in your initial post:

1. What does a negative value represent in my dataset?
2. Find the mean and standard deviation of this data set. Use the following calculator to help find descriptive statistics: https://www.calculatorsoup.com/calculators/statistics/descriptivestatistics.phpLinks to an external site.
3. Test the claim using a hypothesis test at the = 0.1 level. Write out the hypotheses, compute your T value, and make your conclusion based on your results.
4. What are some other variables that may have impacted results?

Supporting Resources: https://www.youtube.com/watch?v=DlwOTOydeykLinks to an external site.

https://www.youtube.com/watch?v=DEkPZv5ppHILinks to an external site.