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## Statistics and probability

### Course: Statistics and probability>Unit 11

Lesson 4: More confidence interval videos

# Small sample size confidence intervals

Constructing small sample size confidence intervals using t-distributions. Created by Sal Khan.

## Want to join the conversation?

• In this series of videos, I don't think Sal explains why for n<30 we should use the t-distribution. Where does the magic number 30 come from? Also, shouldn't the sample size that approximates a normal distribution depend on the population size? •  Positing that the sample distribution adheres to a normal distribution is an assumption. In small data sets, that isn't necessarily true. The t-distribution (also known as the Student t-distribution) is the correction to the normal for small sample sizes. The bigger tails indicate the higher frequency of outliers which come with a small data set. Although as the sample size, n, increases, the t-distribution approaches the normal distribution. At n = 30, the distributions are practically the same, and hence we can use the normal distribution. See the graphical demonstration at the wiki page: http://en.wikipedia.org/wiki/Student%27s_t-distribution, it helps provide the intuition.
• What have I missed? you have a 95% chance of being between 1.4 and 3.3 - but two of the values used to calculate that is outside that inteval (0.9 and 3.9). About 29% of the data set is outside the 95% confidence interval. •  It's not that there's a 95% chance that any sample will be between 1.4 and 3.3, but that there's a 95% chance that the mean of any group of samples will be in that range; individual samples may well be outside that range, dependent on the sample variance.
• Isn't it incorrect to say "There is a 95% CHANCE than that the true value of mu is within...."? It's not a 95% chance... mu is either in the range we calculate, or it's not. Wouldn't it be more accurate to say "We can say with 95% confidence that....." • From Wikipedia article on 'confidence interval':
"A 95% confidence interval does not mean that for a given realised interval calculated from sample data there is a 95% probability the population parameter lies within the interval, nor that there is a 95% probability that the interval covers the population parameter. Once an experiment is done and an interval calculated, this interval either covers the parameter value or it does not, it is no longer a matter of probability. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval.
At , Sal says "There is a 95% chance that our random sampling mean is within 0.96 of the population mean." What he should say is that if this procedure were repeated many times, the results would tend toward this interval with a probability of 95%.
• I cannot grasp how there is a '95% chance mu is within +/- 0,96 of 2,34'. For example if mu is 1,05 then 2,34 is not within 0,96 of mu... I can understand that there is '95% chance 2,34 is within 0,96 of mu' but the logic behind reversing this statement is not clear to me. • If there was a 95% probability that a given interval around mu (we don't know mu, but it has some particular value) contains our sample mean (which we know), then wouldn't there also be the same probability that the same interval around our sample mean contains mu?
(Does mu change because we change our statement? They still have the same relationship to each other and our probability is still the same.)
(1 vote)
• The sample standard deviation is quoted to be 1.04 which is the answer you get if you plug the numbers in excel and apply the formula. However, if you crunch the numbers manually you get 1.08. Anyone aware of why this discrepancy exists? I have done it several times and can't find errors in my calculations. • Why is the population mean is equal to the sample mean? • so i have a problem i can't quite figure out. the problem is as follows:
you randomly choose 16 unfurnished one-bedroom apartments from a large number of advertisements in your local newspaper. You calculate that their mean monthly rent is \$613 and their standard deviation is \$96. What is the standard error of the mean? What are the degrees of freedom for a one sample t statistic?
standard error is the mean/square root of n, or in this case 16, right? which comes out to 24. (69/sqrt 16)
In my book it says to get the one sample t statistic, you take x-bar minus mu divided by the standard error.. but i don't know mu. How do i get mu from just one sample mean? • Where does the t-table come from? I understand the z-table comes from the definate integral of the normal distribution function but how is the t-distribution defined and why is it that small sample sizes tend to follow a t-distribution model rather than some other model? • We use the sample mean and standard deviation as an estimate of the population standard deviation. What do we do when we do a number of trials that each generate 7 data points? How do we estimate the mean and standard deviation in that case?  