Now let's look at what information we can gather from uh diagram that they give us about a distribution here. They give us the diagram of the distribution of the sample means, notice the notation down here saying that it's experts distribution. So it's the sampling distribution of the sample means, oh first, what they're asking us for is what is the mean of the distribution of the sample means? Well from our diagram we see the 12 is right underneath the highest point of the curve, And so 12 is the population mean of our um sampling distribution of the sample means next, it wants us to give them the standard deviation of the distribution of the sample means. So here, from our sampling distribution, we see that this 12.05 is underneath where the curvature changes at a transition point, And also the 11.95 is at a transition point. And those values that are one transition point away from the mean is one standard deviation away from the mean. So we can find our standard deviation of our Sampling distribution by taking the 12.05 and subtracting the mean of 12.00 And we get 0.05, So are stigmas of x bar is 0.05. Now, if n is equal to nine, what is likely true about this shape of the distribution of the population? The population that we grab the samples out of? Well, notice from this diagram, they gave us that diagram is the diagram of a normal shape and it's from the sampling distribution of the X bars. Well, the sampling distribution of expire will have a normal shape if either it was, the samples were taken from a normal distribution to start with. If the sample size is small, Um if we had large sample size, then it population could have a shape that might not be normal but notice they're saying and is equal to nine. Remember nine is considered a small sample size, it's not considered a large sample size until your sample sizes 30 or more. So since an equal nine sample size is small and the distribution of X bar is approximately normal, that means that it's likely true that the distribution of the population had a shape that was normal. Yeah. And then lastly, if an equal nine find sigma, the population standard deviation. Well remember that sigma's of X. Bar can be found by taking sigma divided by the square root of N. We don't know sigma but we know sigma Cybex far from earlier in the problem that 0.5 And we know n. n is nine. So we have 0.5 is equal to sigma divided by the square root of nine. Well the squared of nine is three, so when I saw this equation, I'll just multiply both sides by three to solve for sigma. And we get the sigma is equal to, well, three times 0.50 point 15 Yeah.