Explain how to use a t-test to test a hypothesized mean mu whe...

Question

Explain how to use a t-test to test a hypothesized mean mu when sigma is unknown. What assumptions are necessary?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A researcher wants to test whether the average number of daily steps taken by office workers differs from their recommended 10,000 steps per day. She collects a random sample of 15 workers and finds a sample mean of 9200 steps with a sample standard deviation of 1100 steps. Which type of test is appropriate in this case and what assumptions must be satisfied for it to be valid? Awesome. So it appears for this particular problem we're asked to solve for two separate answers. We're asked to determine which type of test is appropriate to use in the particular conditions set to us by the problem for this particular problem. So we're trying to figure out what type of test is appropriate for this particular case based on the conditions that are provided to us for this specific problem. So what type of test is being used? That's our first answer we're trying to solve for. Our second answer is we're asked to determine what assumptions must be satisfied in order for it to be valid. So what assumptions must we consider that have to be satisfied in order for this particular test that we choose to be considered valid for this particular prompt. So now that we know what we're ultimately trying to solve for. Let's read off our multiple choice answers to see what our final answer might be. A is sign test, population must be binomial. B is 1 sample Z test. Population must be normally distributed. C is 1 sample T test. Sample must be random, independent, and from a normal population. And D is paired T test sample must be paired and large. Awesome. So, with that in mind, our first step is we need to determine the type of test that is being used for this particular prompt. And with that in mind, we need to note and recall that we are comparing a sample mean to a population value which is 10,000 steps in this case due to the fact that since the population standard deviation is unknown to us, the sample size is small, since we have a value of N equals 15. And the test is for whether the mean differs, meaning a two-tailed test is being It differs, meaning a two-tail test in that condition. So that will mean Based on this particular information that we have determined that the appropriate test will be a one sample T test. So with that in mind, so let's write that down. So this is a one. Sample T test. Fantastic. So once again, one sample tea test. So moving right along here, our second step now is we need to create or rather make our assumptions for the one sample T test. So as we should recall, in order to use the T test, the following assumptions must be satisfied. Firstly, random sampling, so I'll call this RS for short. So for random sampling. The data must come from a random sample, as you should recall. Our second condition is independence, which I'll call I for short. The observations must be independent from each other, independent of each other. So, our third assumption we need to prove is the normality, which I'll call N for short. The population of daily steps should be approximately normally distributed, especially important for small samples when N is less than 30. Awesome. So moving right along here. So if the sample size were larger, The central limit theorem would reduce this particular concern. So with that in mind, for our conclusion, A one sample test is appropriate, so we can put a check mark, so say yes. So a one sample T test is appropriate in this particular case because the assumptions are The sample is random, the data is independent. The population, or rather I should say the, yeah, so the population of daily steps is approximately normal. So once again, the population of daily steps is approximately normal. So, with that in mind, that would mean that without a doubt, our final answer will have to be multiple choice answer letter C. One sample T test sample must be random, independent, and from a normal population, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Explain how to use a t-test to test a hypothesized mean mu when sigma is unknown. What assumptions are necessary?

Key Concepts

t-test

hypothesized mean (mu)

assumptions of the t-test

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