Conditions for a t test about a mean | AP Statistics | Khan Academy

Sunil and his friends have been using a group messaging app for over a year to chat with each other. He suspects that, on average, they send each other more than 100 messages per day. Sunil takes a random sample of seven days from their chat history and records how many messages were sent on those days. The sample data are strongly skewed to the right, with a mean of 125 messages and a standard deviation of 44 messages. He wants to use these sample data to conduct a t-test about the mean.

Which conditions for performing this type of significance test have been met? So let's just think about what's going on here. Sunil might have some type of a null hypothesis. Maybe he got this hundred; maybe he read a magazine article that says that, on average, the average teenager sends a hundred text messages per day. So maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which is signified by mu, maybe the null is 100—that they're no different than all other teenagers.

And maybe he suspects, and actually they say it right over here, his alternative hypothesis would be what he suspects: that they send more than 100 text messages per day. So what he does is he takes a sample from the population of days. There are over 365; they say they've been using the group messaging app for over a year, and he takes seven of those days. So n is equal to seven, and from that he calculates sample statistics. He calculates the sample mean, which is trying to estimate the true population mean right over here, and he also is able to calculate a sample standard deviation.

What you do in a significance test is you say, "Well, what is the probability of getting this sample mean or something even more extreme, assuming the null hypothesis?" If that probability is below a preset threshold, then you would reject the null hypothesis, and it would suggest the alternative. However, in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test.

The first is that this is truly a random sample, and that's known as the random condition. You have seen this before when we did significance tests with proportions; here we’re doing it with means: population mean, sample mean. In the past, we did it with population proportion and sample proportion. Well, the random condition says it right here: Sunil takes a random sample of seven days from their chat history. They don't say how he did it, but we'll just take their word for it that it was a random sample.

The next condition is sometimes known as the independence condition, and that’s that the individual observations in our sample are roughly independent. One way that they would be independent for sure is if Sunil is sampling with replacement. They don't say that, but another condition—so you either could have replacement, sampling with replacement—or another way where you could feel that it's roughly independent is if your sample size is less than or equal to ten percent of the population.

Now, in this situation, he took a sample size of seven, and then the population of days says they’ve been using the group messaging app for over a year. So they’ve been using it for over 365 days. Thus, 7 is for sure less than or equal to 10 percent of 365, which would be 36.5. So, we meet this condition, which allows us to meet the independence condition.

Now, the last condition is often known as the normal condition. This is to feel good that the sampling distribution of the sample means, right over here, is approximately normal. This is going to be a little bit different than what we saw with significance tests when we dealt with proportions. There are a few ways to feel good that the sampling distribution of the sample means is normal. One is if the underlying parent population is normal. They don’t tell us anything that there’s actually a normal distribution for the amount of time that they spend on a given day, so we don’t know this one for sure.

But sometimes you might. Another way is to feel good that our sample size is greater than or equal to 30, and this comes from the central limit theorem that then our sampling distribution is going to be roughly normal. But we see very clearly our sample size is not greater than or equal to 30, so we don't meet that constraint either.

Now, the third way that we could feel good that our sampling distribution of our sample mean is roughly normal is if our sample is symmetric and there are no outliers or maybe even say no significant outliers. Now, is this the case? Well, it says right over here the sample data are strongly skewed to the right, with a mean of 125 messages and a standard deviation of 44 messages. So, this strongly skewed to the right, it’s clearly not a symmetric sample data, and so we don’t meet any of these sub-conditions for the normal condition. Thus, we do not meet the normal condition for our significance test.