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Reality often differs from theory; in the real world we seldom know what the true population standard deviation is. The standard deviation refers to the variability of individual observations around their mean. But for this lesson we are going to pretend that the population standard deviation denoted by the symbol sigma, is known to us and we are going to use that to help us construct the confidence interval for the population mean, which itself is denoted by the symbol mu.
A confidence interval is a range of values that expresses the uncertainty associated with a parameter, like the population mean.
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Cases Where This Applies
There are three possible cases where this can be applied.
Case 1: the population standard deviation is known. The sample size is small (n<30). In other words, n, the sample size, is less than 30. And, the population is normally distributed.
Case II: Again, the population standard deviation is known. But, this time around, the sample size is large (n>=30). This means that n is greater than or equal to 30.
In Case III: Again, the population standard deviation is known. The sample size is small (n<30). And, the population is not normally distributed or we don't know its distribution.
The third case uses nonparametric methods to find the confidence interval for mu, meaning we use inferential methods that are not concerned with parameters- such as the population mean or population standard deviation.
Calculating the Confidence Interval
In this lesson, we are going to focus on the first two cases where we use the normal distribution to make the confidence interval for mu. In the first two cases, we would calculate the confidence interval for mu using the following equations:
- Where x bar denotes the value of the sample mean
- Sigma refers to the population standard deviation
- And n refers to the sample size
The value for z is found from standard normal distribution tables for a given confidence level right here. The quantity of z times sigma x bar is the margin of error and it is denoted by the symbol E. In other words, E = z times (x) sigma x bar
Simply put, the margin of error (E) is the quantity we subtract or add to x bar to obtain a confidence interval for mu. Let's build on this to solidify your knowledge of all this crazy terminology with an actual example.
A tech company has just come out with a new cell phone. It needs to figure out the price at which to sell this phone by first figuring out the average price of all similar cell phones available on the market. The company's market research department takes a sample of 16 comparable cell phones to find they have a mean price of $500. The market research department also knows that the population standard deviation of the prices for all such cell phones is $100. Assume the population is normally distributed. Construct a 90% confidence interval for the mean price for all similar cell phones.
First, let's just figure out what we know. We know that the sample size, n = 16. The sample mean, x bar is $500. And the population standard deviation (sigma) is $100. The standard deviation of x bar is simply sigma, divided by the square root of n, using the equation shown before. In our case, that's simply 100/4, which is equal to $25. Using the tables right here, you'd find the value for z for a confidence level of 90% is 1.65. At this point, you have all the values you possibly need to figure out the appropriate confidence interval.
Remember our equations from before; the 90% confidence interval for mu is equal to x bar + - z times sigma sub x bar. Just plug and chug to get 500 + - 1.65 (25). That's equal to 500 + - 41.25. So, we get $458.75 to $541.25. This is our confidence interval. In other words, we are 90% confident that the mean price of all such cell phones is between $458.75 and $541.25.
Now you know how to construct confidence intervals from normal populations when the population standard deviation is known. The standard deviation is the variability of individual observations around their mean. The population standard deviation in our equations was denoted by the symbol sigma, while the population mean was denoted by the symbol mu. The margin of error (E) is the quantity we subtract or add to x bar to obtain a confidence interval for mu.
Using the equations we went over, you should now be able to use them and the tables on this page to construct confidence intervals in the two cases we went over.
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Finding Confidence Intervals with the Normal Distribution
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