The Old Farmer’s Almanac reports that the average person uses 123 gallons of water daily. If the standard deviation is 21 gallons, find the probability that the mean of a randomly selected sample of 15 people will be between 120 and 126 gallons. Assume the variable is normally distributed.
Let's first write down what we know. The population, meaning you is equal to 1 23 The population standard deviation Sigma is equal to 21 and the sample size and is equal to 15. Now we want to know the probability that the sample mean is in between 1 2126 In order to do so, we have to use the central limit zero, which says that the sampling distribution of the sample mean is approximately normal with the mean off new in the standard deviation of sigma over radical. So using that information, we can standardize thes two values and obtain to Z scores and find the corresponding probability values of those e scores and ultimately find the probability that the sample mean is in between these two values. So that was a lot, but let's take a step by step. First, let's find the Z score of 1 2126 to do that, we're gonna use this formula were the Z score is equal to the sample mean minus the population mean over the population standard deviation divided by radical and which is a sample size. So disease four for when the samples sample mean is 1 20 is 1 20 minus 1 23 over 21 divided by radical 15 and that is equal to negative point 55 in the Z score, when the sample mean is equal to 1 26 is equal to 1 26 minus 1 23 over 21 divided by radical 15 that is equal to point fry five. Now, if you use a Z score table, you'll realize that the probability value corresponding to negative 0.55 is equal to point to nine. We want to the probability value corresponding to the Z score 0.55 is equal 2.7088 Now you have to remember when you're reading the values given in a Z score table, it gives you the value left to that Z score. So what do we need by that? Let's first try to understand it visually, So our first see score is negative 0.55 which means that it's less than the average and it's corresponding probability value of 0.29 12 Is Ariel left of it? So it's this red area over here now, our second Z's war is 0.55 and it's a positive value, which means that it's above the average. Okay, it's a 0.553 here and it has a corresponding probability value of 0.7088 And again, this is the area left to this this see score, which is this green area shaded area over here. Now we want to know the probability that the sample mean is in between 1 2126 And we have to remember that the four first Z score is the stent re standardize the sample mean value of 1 20 to get this first see score, and we standardize the sample mean value of 1 26 to get the second Z score. So we have to realize that the probability that the sample mean is in between these two values is just the area in between these two z scores. This blue area that I shaded right over here. So in order to find the blue area, we just have to subtract thes two values, which is 0.7088 minus. That's a minus line 0.2912 and that is equal to point 4176 so we can conclude that the probability that the sample mean is in between 1 20