The average price of a gallon of unleaded regular gasoline was reported to be $\$ 2.34$ in northern Kentucky (The Cincinnati Enquirer, January $21,2006 ) .$ Use this price as the population mean, and assume the population standard deviation is $\$ .20 .$
a. What is the probability that the mean price for a sample of 30 service stations is within $\$ .03$ of the population mean?
b. What is the probability that the mean price for a sample of 50 service stations is within S. 03 of the population mean?
c. What is the probability that the mean price for a sample of 100 service stations is within $\$ .03$ of the population mean?
d. Which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to have at least a. 95 probability that the sample mean is within $\$ .03$ of the population mean?
All right, so we're given the mean and standard error of all gas prices in northern Kentucky. I live in Southern California, and I will kill for 2 34 per gallon. But that's besides the point anyway. So we're given three sample sizes and were asked to find the probability that our sample mean is within 0.3 of our population means So let's start by finding our standard error for this sampling distribution where the sample size equals 30. So, uh, that should not be And bar that should be ex bar. There we go. But anyway, this is standard error over route and so 0.20 over Route 30 and that is zero point 0365 Now we're gonna z score plus and minus 0.3 So oh, rather working to z score an error of plus or minus 0.3 which is an important distinction anyway. Then there's the upper is going to be all right. These are equal to negative 0.82 and positive they're pointed to, respectively. This becomes lower probability, become zero point 2061 and the upper probability because 0.7939 our probability total is our upper probability. Minus are lower probability. So 0.7939 minus 0.2061 That equals 0.5878 All right, part B now an equals 50. We're still finding the same probability. So that's the probability that the, um, sample mean is within 0.3 of our population mean anyway, So let's find our standard error for this sample distribution. Same formula. But it, instead of an equals 30 and equals 50. Oh, you know, though. Yeah. So there we go. This equals zero point 02 83 We're going to do more Z scores with our new standard error. All right. When we compute these, we get negative 1.6 and 1.6 respectively. And then we look at our table. We get that, the our probability lower equals 0.1446 and our upper probability equals zero point 8554 That means our probability. Still, p u minus Pl, which is 0.85 54 minus 0.1446 subtract that out. You get that? That equals 0.71 08 All right, part. See, Now our sample size is 100 and finding the same probability. All right, so our new sample mean are sorry. Sample standard error. It's gonna be 0.20 over the square root of 100 square root of 100 is 10. So this just becomes zero point 0 to 0. Just move the decimal two left, one place. Now, let's see. Score. Once again. Um, all right, 0.20 then see you equals positive. 0.3 over the 0.20 All right, this produces down to negative three over two. So this is negative. 1.50 And this is positive. 1.50 by the same logic. All right, this gives us a probability. Lower of 0.668 probability upper of 0.9332 Our probability equals, uh, p u minus pl So that's 0.9332 when 0.668 That equals 0.8 664 party. Asks us if any of these sample sizes give us a 90% probability that our sample mean will be within boy 03 of our population mean well, if we look back at our answers, 0.5878 is less than 95%. 950.7108 is less than 95% and 0.8864 is less than 95%. So, no, none of these give us 95% probability of a sample mean within 0.3 of our population beat, and there you have it.