Ages of Moviegoers Find the sample size needed to estimate the...

Question

Ages of Moviegoers Find the sample size needed to estimate the mean age of movie patrons, given that we want 98% confidence that the sample mean is within 1.5 years of the population mean. Assume that sigma=19.6 years, based on a previous report from the Motion Picture Association of America. Could the sample be obtained from one movie at one theater?

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A health organization wants to estimate the average cholesterol level of adults in a city. They require 99% confidence that the sample mean is within 4 mg per deciliter of the population mean. Prior data indicates that the population standard deviation is 22.5 mg per deciliter. What is the minimum sample size that is needed? Would sampling only from one clinic be appropriate? Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answers we're asked to ultimately solve for what is the minimum sample size that is needed to accomplish this goal that this health organization wants to achieve, based on the information that is provided to us by the problem itself. And our second answer we're trying to solve for is we're trying to determine would sampling from only one clinic be appropriate to accomplish this specific goal. So with that said, now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is 110, and sampling only from one clinic is not appropriate. B is 110, and sampling only from one clinic is appropriate. C is 210. And sampling from only one clinic is appropriate, and D is 210, and sampling only from one clinic is not appropriate. Awesome. So with that said, First off, in order to solve for this particular problem, let us write down all of our given or all of our known variables. So let us note that our confidence level, which I'll call CL, our confidence level is equal to 99%. So that will mean that capital Z is equal to 2.576, where Z is our Z score value. And our population standard deviation we'll call this value sigma, and sigma is equal to 22.5. And no, keeping in mind, I'm not gonna write our units, but we're told that the units are milligrams per deciliter, and this value is given to us by the problem itself, and we're also told that the margin of error, which we'll call capital E, is equal to 4. Awesome. So now we need to recall a note the following formula. So we need to note that N. Which specifically lowercase n is equal to parentheses Z multiplied by sigma, divided by E all to the power of 2, and when we plug in our known variables, we will find that lowercase n is equal to parentheses. 2.576 multiplied by 22.5. Divided by 4, all to the power of 2, which is going to be, when we plug this into our calculator, it's going to be approximately equal to 210 when we round to one decimal place. So that means that the minimum sample size that is needed will be 210, and that is our first answer that we're trying to solve for. Hooray, we did it, so we solved for one of our final answers. So now we need to determine if sampling from one clinic is sufficient or not. So as we should note and recall, sampling only from one clinic is not appropriate, and it's not appropriate because it may not represent the entire adult population of the city due to potential biases, for example, socioeconomic status, health conditions, location, etc. A more representative or more representative sample should include adults from multiple clinics or a random selection spread across the city. So with that said, our final answer without a doubt has to be multiple choice answer letter D, which states 210 and sampling only from one clinic is not appropriate. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Sample Size Calculation

Confidence Interval

Population Standard Deviation

Watch next