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Confidence Intervals Using the T-Distribution

Confidence Intervals Using the T-Distribution

Author: Ryan Backman
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Calculate a confidence interval using the t-distribution.

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Hi. This tutorial covers confidence intervals using the t-distribution. So, first, let's review the definitions of the confidence interval and the t-distribution. So a confidence interval is an estimate found by using a sample statistic and adding and subtracting an amount corresponding to how confident we are that the interval captures the parameter. And a t-distribution is a distribution similar to the normal distribution but depending on sample size does not diminish towards the tails as fast.

So finding a confidence interval involves four similar steps to carrying out a hypothesis test. So we're going to actually-- we're going to go through the confidence interval steps first, and then we'll take a look at an example and apply each of these steps. So step 1, pick a t star. So this is what's called a critical t-value that matches the confidence level and sample size. Step 2, we're going to check the conditions.

Step 3, calculate the interval using the following formula. It's x bar plus or minus t star, again, that's that critical t-value, times the sample standard deviation divided by the square root of n. Remember that this s over the square root of n is a measure of the standard error. And step 4, state a conclusion in context.

So we're going to apply each of those four steps to an example. So suppose you're interested in estimating the mean amount of money spent on groceries per month for a US household of two adults and two children. So based on a random sample of n equals 125 households, x bar equals $116 and s, the sample standard deviation, is $25 for monthly groceries purchased.

So for these 125 households, on average, $116 was spent per month with the standard deviation of $25. So estimate the mean money spent on groceries using a 95% confidence level. So let's go through this step by step. So we're going to start by picking a value of t star that matches the confidence level and the sample size. So remember that we are dealing with a 95% confidence level, and n equals 125. So we had 125 household sample size.

So to pick this value of t star, I'm going to use my calculator and use a calculator function called inverse t. Again, we're looking for a t-value. So we need a value from the t-distribution. So we do that getting this inverse t. And then your two arguments, the first argument is the tail probability of the confidence level. So the confidence level is 95%, so we're talking about the middle 95%. So that means there's 2.5% on each side. So my tail probability is 0.025.

And then I'm going to do comma and then comma the degrees of freedom. And degrees of freedom is based on the sample size. There's always n minus 1 degrees of freedom for a sample size of n. So in this case, our degrees of freedom is 125. And I'm going to hit Enter there. And it's going to give me a critical t-value of negative 1.979. And when we're dealing with t star, we usually just use the positive version of that critical value, so positive 1.979.

Now step 2, we are going to check the conditions. Again with your conditions, the conditions here are the same as if you were dealing with a hypothesis test. So we need a random sample. We need independent observations. And we need a normal sampling distribution.

So in this case, all of those conditions will be met. We'll assume we are dealing with a random sample. The sample observations were independent, so we sampled less than 10% of the population size. And we have a normal sampling distribution because n is 125, which satisfies the central limit theorem.

Step three, now we're actually going to calculate the interval using the formula. So we're going to start with x bar. So x bar was 116. And we're going to go plus or minus our critical t value, which was 1.979. And we're going to multiply by our sample standard deviation divided by our sample size. So our sample standard deviation was 25. And I'm going to divide by the square root of 125.

So I think what I'll do first is I'm going to do the whole margin of error in one step on my calculator. So what I'm going to do is take that critical t-value, 1.979, and I'm going to multiply it by the standard error, so 25 over the square root of 125. And hit Enter there. So I'm going to get a margin of error of about 4.425. So not I have 116 plus or minus 4.425.

And then to write it actually as an interval instead of as a point estimate plus or minus a margin of error, I'm going to actually do 116 minus the margin of error and 116 plus the margin of error. So I'm just going to do that on my calculator. 116 minus my last answer.

So that's going to be about 111 point-- and then I'm dealing with money here, so I'm going to round this to the nearest penny. And let's do the same thing here. I'm just going to call up my margin of error again in the new 116 plus the margin of error. And that's about $120.43. And again, the units on here are dollars.

And we are now going to move to step 4, state a conclusion in context. So what I'm going to say now is that I am 95% confident-- so I'm going to quote that confidence level-- that the interval $111.57 to $125.43 captures the mean monthly grocery bill for a US family of four.

So again, the confidence interval tells me that I'm 95% confident that this interval captures that true population parameter. So that is step four of your confidence interval procedure. So that covers confidence intervals using a t-distribution. Thanks for watching.

Additional Practice Problems

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Terms to Know
Confidence Interval

An interval we are some percent certain (eg 90%, 95%, or 99%) will contain the population parameter, given the value of our sample statistic.

t-distribution

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.

Formulas to Know
Confidence Interval of Population Mean

C I space equals space x with bar on top space plus-or-minus space t asterisk times space bevelled fraction numerator s over denominator square root of n end fraction