This tutorial will discuss confidence intervals, specifically that use the t-distribution. The primary focus of this lesson is:
These are going to be confidence intervals for population means. Take a look at an example.
Many times, consumers will pay attention to nutritional contents on packaged food. So it's important to make them accurate as to what their contents actually contain. A random sample of 12 frozen dinners was selected, and the calorie contents of each one was determined. The stated calorie content was 240.
One of the boxes contained actually 255 calories worth of food whereas another one only contained 225 calories worth of food.
You want to construct a 90% confidence interval for the true mean number of calories. You want to construct a confidence interval such that you’re 90% confident that the true mean of all the packaged frozen dinners lies within the interval. Doing a confidence interval is a lot like doing a hypothesis test, and there are a lot of the same requirements. First, verify the conditions for inference are met. Then, calculate the confidence interval and, finally, interpret it in the context of the problem.
Look at the conditions for this problem. The randomness condition.
How are the data calculated? It was a random sample, as was said in the problem. How about the independence condition? Is the population of all frozen
dinners at least 10 times the size of your sample? That's reasonable to believe. Assume there are at least 120 frozen dinners in all of this company's frozen dinner line. Fnally, the normality condition. This one's a little tricky. Your sample size isn't 30 or larger, so the central limit theorem doesn't apply to this problem. Is the parent distribution normal? You don't know that either. You need to determine if this is plausible. You can do that by graphing the actual data that you have.
You can see that the parent distribution might be normal since the data that you got from the population are single peaked and approximately symmetric. It's possible that the population parent distribution is normal. You can proceed under the assumption of normality. You can’t verify it 100%, but go with it for the purposes of this problem.
Next, calculate the confidence interval. Normally you would take the sample mean plus or minus some number of standard deviations times the standard error, because this is a sampling distribution.
The only problem is this formula has a sigma in it. And we don't know what the population standard deviation is. You have to replace this formula with one that uses s. Since you're using s as a stand-in for sigma, you need to use the t-distribution instead.
Confidence Interval of Population Mean
You have this information from our sample.
n is the sample mean when you calculated the mean of the 12 dinners. There were 12 of them. This was the mean. And s was the sample standard deviation.
What you need to do is figure out what that t star value is going to be.
If you look closely at this model above, this is a t-distribution. It says t star. That means you need a t that will give us 90% of the t-distribution. One way to do it is to look and see the upper tail probability would be 0.05, because then there would be a lower tail probability that was also 0.05. That would give us 90% in the middle. Or look all the way down at the bottom and see that there is a row way down here that says confidence level c. And there's 50%, 60%, 70%, 80%, 90%.
Either one of those justifications is reason enough to use this column. Next, you need to know is which number from this column you’re going to use. Check is the degrees of freedom row.
In this problem, you had 11 degrees of freedom because you had 12 dinners in your sample, and the degrees of freedom is n minus 1. Look in the 11 degrees of freedom row and the 90% confidence column until we obtain a t star of 1.796.
Now you have all the information needed in order to create your confidence interval. Construct it as x bar plus or minus the t critical value times the sample standard deviation divided by the square root of sample size.
When you do that, you obtain 244.33 plus or minus 6.65 is the result of all this multiplying and dividing here. When you subtract and then add, you have 237.68 for the lower bound and 249.98 for the upper bound.
What does this confidence interval actually mean? How can you interpret the interval? The interpretation is that we're 90% confident that the true mean calorie content of all frozen dinners is between about 237 and 250 calories. We're 90% confident that the real value is somewhere in there, and that 240 value that they were purporting at the beginning of the problem is, in fact, plausible.
We can create point estimates for the population means using x bar, and determine
the margin of error. That margin of error is the t star times s over the square root of n piece of the confidence interval. First, we verify that conditions are met. Then we construct and interpret the confidence interval. So we talked about confidence intervals specifically for means using the t- distribution.
Source: This work adapted from Sophia Author Jonathan Osters.
An interval we are some percent certain (eg 90%, 95%, or 99%) will contain the population parameter, given the value of our sample statistic.
A family of distributions similar to the standard normal distribution, except that they are fatter in the tails, due to the increased variability associated with using the sample standard deviation instead of the population standard deviation in the formula for the test statistic.