A leading magazine like barrons reported at one time that...

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 18 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 18 weeks and that the population standard deviation is 2.7 weeks. Suppose you would like to select a random sample of 80 unemployed individuals for a follow-up study. Find the probability that a single randomly selected value is greater than 17.8. P(X > 17.8) = (Enter your answers as numbers accurate to 4 decimal places.) Find the probability that a sample of size n=80n=80 is randomly selected with a mean greater than 17.8. P(X̄ > 17.8) = (Enter your answers as numbers accurate to 4 decimal places.)

Transcript

00:01 We're looking at a distribution here, and we want the probability that a random value is greater than 17 .8.

00:10 So we know that the mean, mu, is 18, and we know that the standard deviation is 2 .7.

00:18 Okay, so to work out the probability that a randomly selected value is greater than 17 .8, we're going to have to assume that this is normally distributed.

00:29 So let's draw a normal distribution and look for that.

00:36 With a normal distribution, we don't use raw data, we need to standardize it.

00:40 So we turn it into a z score, and z is equal to x minus mu over sigma.

00:46 So i call this part a.

00:48 So we have 17 .8, minus 18, divided by 2 .7, which gives me a z score of minus 0 .074.

01:01 And this actually is recurring.

01:05 So i'm keeping this in my calculator from the next step, and that's just below the mean, very slightly below the mean, and we want greater than, so we'll be area to the right.

01:18 Now how do we turn this into a probability? well, we need either a z score table, a graphical calculator, or something like excel.

01:26 Recommend against the table because it limits how many decimal places you can put in.

01:30 Sometimes you're told to use one, but if not, prefer technology.

01:34 For more accuracy.

01:36 So, calculator or excel or something else, there are two functions that will take you from z score to probability.

01:44 The standard and the cumulative functions.

01:47 The standard function, when you put in your value, gives you the area between your cutoff and the mean.

01:52 Not entirely what you want, but total area under this curve is one, and it's symmetric, so 0 .5 to the right to the mean, 0 .5 to the left.

02:02 So if you take this and add, you get the entire green area.

02:08 Cumulative function gives you the area to the left.

02:11 So this in red, also not what you want, but take it away from one, and you'll have the area to the right.

02:17 I'll use the standard function, so whatever i get, i need to add 0 .5 to.

02:23 Okay, so i put my z score in the normal function, and it gives me 0 .0 .295, so my answer is 0 .5 -295 to 4 decimal places.

02:36 That's the area of this and the probability, but a randomly selected value falls into that area.

02:44 Okay, part b...

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