Lets say that this is a data set of a random sample of the time a person spends on social media in a day in minutes. The mean of this data set is 144,
and from the mean of this data set, we could infer that the average or mean of the population is also 144. This is called a point estimate,
and a point estimate is a single value estimate for a population parameter. A parameter is a characteristic of a population and a statistic is a characteristic
of a sample. So, basically we use statistics to make inferences and predictions about populations. However, the probability that the actual population mean is 144
is virtually zero, so we can use an interval estimate. An interval estimate is a range that is used to estimate a population parameter.
For instance we have our point estimate of 144 here in the middle, and we could say use 10 as a margin of error, so, 134 and 154 here on the line.
We would then say 144 plus or minus 10, or 134 less than mu, less than 154. And, the formula for a confidence interval for the mean is x bar plus or minus,
the margin of error. But, before finding a margin for error, we first need to determine how confident we want to be with our interval estimate.
There are typically 3 common confidence intervals that are used, 90%, 95%, and 99%. We know from the central limit theorem that when the sample size is greater than
or equal to 30, and data set this mean is from has a sample size of 35, the distribution of the sample means is a normal distribution.
Since we know the distribution is normal, and it is a random sample, we can use the standard normal distribution and z scores to calculate confidence intervals
for the mean. So, if we want a 95% confidence interval for a mean, on a standard normal distribution that looks like this. We have the mean here in the middle,
and 95 divided by 2 is 47.5. So 47.5% on the left side of the mean and 47.5 on the right side. And what we need to find is the value for each of these lines
to determine the values of our confidence interval. We know that the total percentage under the curve is equal to 1. So, 1 minus 95 equals 5%,
which is what is under these two ends or tails. Next, we can divide 5 by 2, and we have 2.5% on the left tail, and 2.5% on the right tail.
Now we can use z scores. A z score table tells us the total area to the left of a particular z score. So, we need to find the z score for 2.5%, the area to the far left.
In the z score table that value is -1.96, as you see here. Since we know the standard normal curve is symmetrical about the mean, we know the other line will have
a z score of positive 1.96. These z scores are critical values, and we can label these as negative z alpha over 2 and positive z alpha over 2,
where alpha is equal to the total area in both tails of the curve. Now back to our confidence interval formula of x bar plus or minus the margin of error. In this formula,
x bar is the sample mean that we got from our data set and the margin of error is equal to plus or minus z alpha over 2 times the population standard deviation
over the square root of n, where n is the sample size. The population standard deviation over the square root of n is called the standard error of the mean,
and this formula can also be written out like this, with mu in the middle. And there are 3 rules for using this method, number 1 is that the sample must be a random sample,
and number 2 is that the sample size n must be greater than or equal to 30, or the population must be normally distributed. And number 3 is that the population
standard deviation must be known. So, lets go through an example. Using the data from earlier, time spent on social media, which has a mean of 144,
the sample size is 35, and we are going to assume the population standard deviation is equal to 61. Find a 90% confidence interval, a 95% confidence interval
and a 99% confidence interval for the population mean. First for 90%, using our formula our interval is x bar, minus z alpha over 2, times the population
standard deviation, divided by the square root of n, less than mu, the population mean, less than x bar plus z alpha over 2, times the population standard deviation,
divided by the square root of n. The first thing we need to do is find the value for z. Since we are looking for a 90% interval, this section here under the curve,
we have 5% in each of the tails here. So, we can find the z score for 5% or .0500 in the z table, which is right in the middle of -1.64, and -1.65,
so we can use a midway value of -1.645. Now we have everything we need, so, z alpha over 2 equals -1.645, and positive 1.645.
Calculating out, we get a margin of error of 16.96, which gives us, 127.04 less than mu, less than 160.96, so we can say with 90% confidence
that the interval between 127.04 and 160.96 minutes does contain the population mean based on a sample of 35 people.
Now for part 2, 95%. The only thing that is going to change from the previous problem is the z score, as this time we are looking for 2.5% under each of the tails.
And we found this z value earlier in the video, which is -1.96 from the z table. Now we can plug this into the formula, and we get a margin of error of 20.21,
which gives us 123.79 less than mu, less than 164.21. And for part 3, 99%, we have 0.5% in each of the tails, so we look up .0050 in the z table
and we get between -2.57 and -2.58, so, we can again use a midway value of 2.575. Plugging in and calculating out, we get a margin of error of 26.55,
which gives us 117.45 less than mu, less than 170.55. So, here are our three intervals, and you can see as the percentage or confidence
goes up, the intervals get larger. Alright my friends, that, that be the basics on the confidence intervals for the population mean when the standard deviation is known.
Aren't you excited? I know, I'm a goofball. Anyway, do have more videos right there for you, till next time, I am outta here.