In a past video we covered how to construct a confidence interval for a population mean when the population standard deviation is known.
Real quick, we use the formula x bar minus z alpha over 2, times the population standard deviation over the square root of n, less than mu,
less than x bar plus z alpha over 2, times the population standard deviation over the square root of n. In this formula, Sigma is the population standard deviation,
which is known. So, to use this formula, we need to know the population standard deviation. However, in most cases, the value of the population standard deviation
is unknown and when it is not known, we use the formula x bar minus t alpha over 2, times s, over the square root of n, less than mu, less than x bar plus t alpha over 2,
times s, over the square root of n. In this formula, x bar is the mean from a sample, s is the standard deviation from a sample, n is the size of the sample,
t alpha over 2 is a t score based on the confidence interval we are using and the degrees of freedom, and the degrees of freedom are equal to n minus 1,
the sample size minus 1, and mew is the population mean, which is unknown. And to use this formula, the sample must be random and the sample size
must be greater than or equal to 30, or the population must be normally distributed. The t distribution is actually a family of curves,
which are determined by the degrees of freedom. For instance here is an example of a t distribution where the degrees of freedom are equal to 2,
and here is a t distribution for 5 degrees of freedom, and here is a t distribution for 20 degrees of freedom. And now here is the standard normal distribution, or z curve.
If we put them together, you can see as the degrees of freedom increase, or as the sample size increases, the t distribution approaches the standard normal distribution,
the z curve. And we will use the symbol d dot f dot, for the degrees of freedom. Just like in the standard normal distribution, or z curve, the t distribution
has an area under the curve equal to one, the tails never touch the x axis, it is symmetric about the mean, and the mean is equal to zero.
Unlike the z curve, which has a standard deviation of 1, the standard deviation for a t distribution varies but is greater than 1. As far as the 90%, 95% or 99%
confidence intervals, they have the same concept for a t distribution. For instance for a 90% confidence interval for a t distribution we have 45% to the left of the mean,
45% to the right of the mean, which is the actual confidence interval, and 5% in each of the tails. And just like with a z score, we would look up the t score
in a t distribution table. And here is a t distribution table, which can be found through an internet search, or in any statistics book, and for confidence intervals,
we will use the top row, under the 90% column. Say our sample size was 24, then the degrees of freedom, n minus 1 would be 24 minus 1, or 23.
So, under column 1, df, we would go down to 23 and over to the 90% column. And we get 1.714, which is the t score. And going back to our formula,
we would plug this value, 1.714 in for t alpha over 2. Lets do an example. So, lets say that you put together a random sample of the time customers spend
having dinner and taking up a table at a fancy restaurant. The sample had a mean of 88 minutes with a standard deviation of 17, and your sample size was 14,
and we are going to assume the population is normally distributed. Find the 95% confidence interval of the population mean for the time spent at dinner.
Using our formula, we have x bar equal to 88, s or the sample standard deviation equal to 17, n, the sample size is equal to 14, and for t alpha over 2,
we have a confidence level of 95%, and the degrees of freedom equals n minus 1, which is 13. In the t distribution table we go down to 13 in the df column,
then over to the 95% confidence interval column. And the value listed is 2.160, so this is our t alpha over 2 value, and our formula is complete.
Plugging and calculating we get 78.2 less than mu less than 97.8. So, we can be 95% confident that the population mean is between 78.2 and 97.8 minutes
based on a sample of 14 customers. Alright my friends, that be the basics on finding a confidence interval for the mean, when you don't know the standard deviation.
Hopefully this video helped you out, I do have more videos right there for you, till next time, I am outta here.