The weights of packets of chips are known to be distributed with a variance of 2.56 grams. A simple of 100 packs were tested and the mean weight was 24.1 grams. Calculate a 99-percent confidence interval for population mean 𝜇.
So we are being asked to calculate a confidence interval. A confidence interval is a range of values we are fairly sure our true values lie in. So if we were asked to calculate a 99-percent confidence interval, that means we have to have a specific value of 𝑍, and this value of 𝑍 can be looked up, and it’s equal to the value of 2.576.
But how do we find this confidence interval? The confidence interval formula is 𝑋 bar plus 𝑍 times 𝜎 divided by the square root of 𝑛. 𝑋 bar stands for the mean, 𝜎 stands for the standard deviation, and 𝑛 stands for the number of observations. So let’s begin looking through the question and find what we are given.
It says the weights of packets of chips are known to be distributed with a variance of 2.56 grams. Variance is equal to 𝜎 squared, and the variance tells us how much our values vary from the mean. So that variance is equal to 𝜎 squared, and standard deviation is equal to 𝜎, to go from variance to standard deviation, we would simply need to square root it.
So if the variance is 2.56 grams, we will square root that and get the standard deviation. Therefore, the standard deviation is equal to 1.6. So we can replace the 𝜎 with 1.6. Next in the question, it says a sample of 100 packs were tested. Therefore, the number of observation would be 100.So we can replace 𝑛 with 100.
And then lastly, it says and the mean weight was 24.1 grams. So we can replace 𝑋 bar with 24.1. And lastly we have 𝑍, and 𝑍 goes with the percent of a confidence interval and we want the 99-percent confidence interval, which use-uses a 𝑍 value of 2.576. And now we need to simplify. The square root of 100 is 10. And now we can take 1.6 divided by 10, which is equal to 0.16. And now we need to take that number and multiply it by 2.576.
And after multiplying, we get 0.41216. Now before adding these numbers together, we are asked to calculate an interval. If we would simply add these numbers together, we won’t get an interval, we’ll simply get one number. So the way that we get an interval is we take the 24.1 and we add our decimal, and then we take the 24.1 and we subtract our decimal.
That will create an interval. So, so if we would add our numbers together, we would get 24.51216. And if we would subtract them, we will get 23.68784. Let’s go ahead and round each of these numbers to two decimal places. So looking at this one, we will either round it up to a two or keep it a one. So looking at the number to the right of that, since it’s less than five, we will keep the one a one.
Next, we need to decide the same thing for the eight, so we look at the number to the right, which is seven. And since it’s five or larger, we will round the eight up to a nine. This means our confidence interval for the population mean 𝜇 is at least 23.69 and at most 24.51. This will be our 99-percent confidence interval.