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### 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?

• 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)
• Why is the population mean is equal to the sample mean?
• It's not. It's what the s a m p l e means are randomly distributed around.
We use this fact to calculate confidence intervals.
(1 vote)
• 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?
• The standard error of the mean is the standard deviation divided by the square root of the sample size, or s/√n. You shouldn't need to calculate a test statistic to find the degrees of freedom.
• 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?
• I have a question, why did Sal mulitply 0.39, the standard deviation of sample distribution, to 2.447 to get the distance from the miu to the critical value?
• He multiplied because 2.447 is the t-critical value that corresponds to a 95% two-sided confidence interval using a t-distribution. The 2.447 is a standardized value that explains what t-values will contain 95% of the t-distribution. The t-critical value must then be converted back to units of the original question. Multiplying by the standard deviation of the sampling distribution will then result in the distance from the sampling mean (mu).

When finding one-sample t confidence intervals, the general equation x_bar +/- (t critical value)*s/sqrt(n) is used. The multiplication is the (t critical value)*s/sqrt(n).
(1 vote)
• Why do you know the population mean is the sampling distribution mean? What's the difference between that mean and the mean of 2.34 in the video?

Sample size of 30 needed to be normal? What if you're at 20 - 29?
• According to something called the Central Limit Theorem in Statistics a sample size of 30 is the minimum needed in order to run a valid statistical test (unless the data appears to be normal through another method such as a Normal Probability Plot). The sample mean or x-bar is the mean of the sample size that the researcher went out and collected. The actual mean is the true population mean. For example, if I wanted to find the true mean height of people. The actual mean would be records provided by the American Medical Association. The x-bar or sample mean would be from the sample I took the data from (ex.42 of my neighbors).

Hope this helps!
(1 vote)
• I've had trouble distinguishing when to use a t-table vs a z-table.

Do we use Z-tables for proportions or sample means when we assume a normal distribution? n>=30 or approximately normally symmetric

Do we use T-tables for any other instance? Because at first, I thought Z-tables were for proportions and T-tables were for means.
(1 vote)
• Z- use if you know the POPULATION standard deviation, sample size >30, AND Independence is known
T - use if one or more of the conditions listed above is broken
T is more robust than Z so it is used more often. Also rarely is population standard deviation is known so z will be used less often.