You wish to estimate the mean winning time for Boston Marathon...

Question

You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.

Accepted Answer

Hello 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. You want to estimate the average time in units of hours it takes hikers to complete a mountain trail. The estimate must be accurate to within 0.033 hours, 2 minutes of the population mean. You decide to construct a 99% confidence interval for the population mean. Use the population standard deviation of 0.068 hours. What is the minimum sample size required? Awesome. So it appears for this particular problem, we're ultimately trying to determine what is the minimum sample size required for this particular prom based on the information that is given to us by the prom itself. So now that we know, once again, we're ultimately trying to figure out what the minimum sample size required to meet the conditions set to us by the prom itself, let's read off our multiple choice answers to see what our final answer might be. So A is 29, B is 26, C is 32, and D is 24. Awesome. So, first off, let us recall once again that we are asked to calculate the minimum sample size, which is the value and we're ultimately trying to solve for the value N, so N is unknown to us. So once again, since we're trying to calculate the minimum sample size, We need to recall that we need, we need to recall and use the data points that are given to us by the problem itself. So we're told that the desired margin of error, which will denote this value as E and E is equal to 0.033 hours. We're told that the population standard deviation, which we can denote this as sigma, and sigma is equal to, as the prom states, 0.068 hours. Fantastic. And once again, our confidence level, which I'll call CL for short, our confidence level is equal to 99%. Fantastic. So now our first step that we need to take is we need to find the Z score for a 99% confidence level. So as you should recall, for a 99% confidence, the critical Z score is. Equal to 2.576, and we can achieve this value by using a standard normal distribution table, which this should be table 4 from your elementary statistics textbook. Specifically the 8th edition, and once again you can use any standard normal distribution table or you can use table 4 from that specific textbook. But when you use your standard normal distribution table, you will determine that for a 99% confidence, the critical Z score will be 2.576. So with that in mind, our next step is we need to recall and use the minimum sample size formula, which states that N is equal to parentheses. The subscript. Alpha divided by 2 to the power of sigma. Awesome. So once again, we're using our minimum sample size formula which states that N is equal to parentheses. Z subscript alpha divided by 2 power of sigma, all divided by E. And our and then we need to take this big expression to the power 2. So once again, we need to take Z subscript alpha divided by 2 to the power of sigma divided by E, all to the power of 2. So, With that in mind, we now need to plug in all of our known variables, and when we plug in all of our known variables, we will find that this is equal to parentheses 2.576 multiplied by 0.068. All divided by 0.033, and don't forget that this is to the power of 2. So that will mean that N is equal to 28.176, which will mean when we round to the nearest whole number. Which it's important to note that since the sample size must be a whole number, we need to round up, and when we round up, you will find that it's approximately equal to 29. So that will mean that our final answer, without a doubt, will have to be multiple choice answer letter A, which is 29, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Sample Size Determination

Standard Deviation

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