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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 11

Lesson 1: Constructing a confidence interval for a population mean- Introduction to t statistics
- Simulation showing value of t statistic
- Conditions for valid t intervals
- Reference: Conditions for inference on a mean
- Conditions for a t interval for a mean
- Example finding critical t value
- Finding the critical value t* for a desired confidence level
- Example constructing a t interval for a mean
- Calculating a t interval for a mean
- Confidence interval for a mean with paired data
- Making a t interval for paired data
- Interpreting a confidence interval for a mean

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# Example constructing a t interval for a mean

Example showing how to calculate a one-sample t interval for a mean.

## Want to join the conversation?

- I calculated the t-score on my Ti-84 by using the invT(area, df) function, and with area=0.95 and df = 13, I got the t-score to be approx.1.761, can someone help? Thanks!(4 votes)
- You need to input the one-tailed area, in this case 0.975(3 votes)

- I used to calculate the critical point over the course of the previous videos by multiplying the value from the table by the square of p/sample n. What's new this time?(1 vote)
- If our df is 60 and we need to do 60-1=59, which one will we choose? Still the 60 or 50?(1 vote)
- We’ve established that heights of 10-year-old boys vary according to a Normal distribution with μ = 140 cm and σ = 5 cm.

What proportion is between 150 and 140 cm?(1 vote) - The weight of brains from Alzheimer cadavers varies according to a Normal distribution with mean 1077g and standard deviation 106g. The weight of an Alzheimer-free brain averages 1250 g. What proportion of brains with Alzheimer disease will weigh more than 1250 g?(1 vote)
- Why do we divide the sample standard deviation by the square root of n? Don't the z* and t* statistic give the number of
*standard deviations*to go away from the mean? If so, then why are we dividing something by the standard deviation? Thanks for the help!(1 vote)- Remember that the sample variance is equal to population variance divided by the sample size. When you square root everything you get the standard dev. You can check out the video "Standard error of the mean" to review.(2 votes)

## Video transcript

- [Instructor] A nutritionist
wants to estimate the average caloric content of the burritos at a popular restaurant. They obtain a random sample of 14 burritos and measure their caloric content. Their sample data are roughly symmetric with a mean of 700 calories and a standard deviation of 50 calories. Based on this sample, which of the following is
a 95% confidence interval for the mean caloric
content of these burritos? So, pause this video and see
if you can figure it out. All right, what's going on here? So, there's a population of burritos here. There is a mean caloric
content that the nutritionist wants to figure out but doesn't know the true population parameter
here, the population mean and so, the take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean which is 700, they also calculate the
sample standard deviation which is equal to 50 and they want to use this data to construct a 95% confidence interval and so, our confidence
interval is going to take the form and we've seen this before, our sample mean plus or
minus our critical value times the sample standard deviation divided by the square root of N. The reason why we're using a T statistic is because we don't know the actual standard
deviation for the population. If we knew the standard
deviation for the population, we would use that instead of
our sample standard deviation and if we use that, if we use sigma which is a population parameter then we could use a Z
statistic right over here, we would use a Z distribution but since we're using this
sample standard deviation, that's why we're using a T statistic but now let's do that. So, what is this going to be? So, our sample mean is 700, they tell us that, so it's going to be 700 plus or minus, so what would be our critical value for a 95% confidence interval? Well, we will just get out our T table and with the T table, remember, you have to care about degrees of freedom and if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is N minus one, so that's gonna be 14 minus one is equal to 13. So, we have 13 degrees of freedom that we have to keep in mind when we look at our T table, so let's look at our T table, so 95% confidence interval and 13 degrees of freedom, so degrees of freedom right over here, so we have 13 degrees of freedom, so that is this row right over here and if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see, I'll draw it really small, a little small distribution
right over here, so if you want 95% of
the area in the middle, that means you have five
percent not shaded in and that's evenly divided on each side, so that means you have
two and a half percent at the tails, two and a half percent, so what you want to look for is the tail probability
of two and a half percent. So, that is this right over here, .025, that two and a half percent and so, there you go, that is our critical value, 2.160. So, this part right over here, so this is going to be two, let me do that in a darker color, this is going to be 2.160 times what's our sample
standard deviation? It's 50 over the square root of N, square root of 14, so all of our choices have the 700 there, so we just need to figure out what our margin of error, this part of it and we could use a calculator for that. Okay, 2.16, ic an write a zero there, it doesn't really matter, times 50 divided by the square root of 14, we get a little bit of a drum
roll here I think, 28.86. So, this part right over here is approximately 28.86. That's our margin of error and we see out of all
of these choices here if we round to the nearest 10th that would be 28.9, so this is approximately 28.9 which is this choice right over here. This was an awfully close one. I guess they're trying to make sure that we're looking at enough digits. So, there we have it. We have established our
95% confidence interval. Now, one thing that we
should keep in mind is is this a valid confidence interval? Did we meet our conditions for
a valid confidence interval? And here we have to think well, did we take a random sample and they tell us that they
obtained a random sample of 14 burritos, so we check that one. Is the sampling
distribution roughly normal? Well, if you take over 30 samples then it would be but here we only took 14 but they do tell us that the sample data is roughly symmetric and so, if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal and then the last condition is the independence condition and here if we aren't
sampling with replacement and it doesn't look like we are, if we're not sampling with replacement, this has to be less than 10%
of the population of burritos and we're assuming that
there's going to be more than 140 burritos that this popular restaurant makes. So, I think we can meet the
independence condition as well, so assuming that you feel
good about constructing a confidence interval, this is the one that you
would actually construct.