Suppose that a category of world class runners are known to...

Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Let $\overline{X}$ the average of the 49 races. a. $\overline{X} \sim$ _____(_____,_____) b. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons. c. Find the $80^{\text { th }}$ percentile for the average of these 49 marathons. d. Find the median of the average running times.

Transcript

00:01 World -class runners, we have the average time to finish the marathon is 145 minutes.

00:06 Standard deviation is 14 minutes.

00:08 We're going to take a sample of 49 of these races.

00:13 The mean of the average or of the, let me rephrase this, the average or the sample distribution means is going to be the same as the population means.

00:29 So this is still 145, whereas the standard deviation of the sample means is going to equal the standard deviation of the entire population divided by the square root of the sample size to square root of 49.

00:49 This comes out to equal square root of 49 is 7, so 14 divided by 7 is 2.

00:55 So for the sample means, where samples are of size 49, okay, we have a sample size 49 races, so each sample has 49 races.

01:09 The average or the mean of the sample means, the mean of the sample mean distribution, it's a mouthful, is 145, and the standard deviation of the sample means is 2.

01:24 Now, if the standard deviation of the sample means is 2, then the variance, sigma squared, is going to be the standard deviation squared or 2 squared is 4.

01:39 So, x, the distribution of the sample means will follow a normal distribution with a mean of 145, and a variance.

01:55 Of 4.

01:57 Now, if we select a sample of 49 races, what is the probability that the mean of our sample, the average running time to finish the race, was between 142 and 146 minutes.

02:14 Now remember, for our distribution of sample means, the mean is 145, standard deviation is 2, so we're going to use the applets set on the probability appellate set on normal distribution.

02:26 The mean is 145, standard deviation is 2.

02:31 Here is a graph of the distribution of the sample means.

02:37 Now, what's the probability that our sample mean? the mean of our sample is between 142 and 146 minutes.

02:45 142 and 146 minutes.

02:58 The probability that the mean of our sample will be between 142 and 146 minutes.

03:04 Minutes is 0 .6247.

03:14 Okay, so what race time represents the 80th percentile in our distribution of sample means? so the probability that x is less than or equal to the 80th percentile amount will be 0 .80.

03:38 So what number of minutes will be the 80th percentile? to the left of a value.

03:47 Now, 50 % will be here.

03:51 95 % would be here, two standard deviations above the means...

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