In this video we discuss how to find confidence intervals for the population mean, from a sample, when the population standard deviation is known and we also cover what is and how to find the margin of error. Transcript/notes (partial)
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 for error, so, 134 and 154 here on the line. We would then say 144 plus or minus 10, or 134 less than mew, 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, the distribution of the sample means is a normal distribution. Since we know the distribution is normal, 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. Z scores tell 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 mew in the middle.
And there are 2 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.
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 part 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 mew, the population mean, less than x bar plus z alpha over 2 times the population standard deviation divided by the square root of n.
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.
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 mew, 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 30 people.
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