QUESTIONWhich of the following is not true for normal distribution? The further value is from the mean; the lower its probability:The mean is equal to the variance_...

What are the values of the mean and standard deviation of a standard normal distribution?

This question asks, What does the mean off a standard normal distribution and what is the standard deviation off? A standard normal distribution for a standard normal distribution which looks something like this you could, which looks something like Thetis. The mean is zeal. The mean is zero and standard deviation. That s sigma is equal toe one mean is equal to zero. Whenever we have really life data sets, we generally try toe scale them such that they're mean become zero and the standard deviation becomes one.

All right. So we're going to evaluate normal distributions and talk a little bit about the correlations between the different proportions of different percentages, sooner deviations and means how they all kind of work together. So first, we're gonna look at what proportion or percentage of a normal distribution would be greater than the mean. So again, a normal distribution is that bell curve. So it has two tails in your mean often called mu. Right here is right in middle. So mean is an average. So that's why it threatened middle there. So in a normal distribution, it's symmetric. So what's on the right hand side of this bell curve on what's on the left hand side are suitable that same. So the mean is smack dab in the middle There. Therefore, on either side of the mean, you're gonna have 50% of your data is gonna be on the positive side of your mean and 50% of the data is going to be on the negative side, and that all would encompass to 100% of your data that would fall within their but based off of this, what would we look at? Will redraw list, so it doesn't get too messy. But based on this, we want to know in general what proportion lies within one standard deviation of the mean. So, based on the impure kal rule, we can look at one standard deviation two and three. And then again on the opposite side Negative one standard deviation negative to a negative three. So what would fall in between this area right here? Or one standard deviation from the mean? And if you look at the empirical rule, it states that 68% of your data should fall in that shaded region, Oregon, one standard deviation. Now, if you were to look at too, you add in here, then you would have 95% of your data would fall in there, and then you shaded even further all the way out to the three standard deviations would be 99.7. Again, I'm not necessarily saying it for that side. It's for everything. So I should probably put that underneath, But, um, you get the picture there and then we also want to evaluate how this would work as a visual. If we're trying to find certain areas, so we'll just get rid of this guy just to keep it click. So let's say we were curious knowing all this data. We wanted to say what proportion would be greater than the value that is one standard deviation below the mean so in a normal distribution again for the data right here, one senior distribution, 68%. What? You should be able to see their I don't see the toolbar at the top, so 68% is within that. And once the inter deviation. So we would want to see you know what would be the proportion that's greater than one standard deviation. So if you have to think about it logically, you have 68% that's already within that right. And then we want to add on greater than so what we would do with that. We would actually find 100% minus the 68% to see the left over, and then you would divide that by two because there's two different tales, and that should come out to 84%. So the proportion that's greater than a value that is one standard deviation below the mean over here below you hear below would be 84%

We're talking about the normal distribution right now in the things that determine its shape. This question gives us two distributions one with a mean of negative four and the standard deviation of three and the other. We'll have a mean of six in a standard deviation of negative of Excuse me of positive three. Now, first it asks, Are these going to have the same shape? Well, which of these parameters determines the shape? We know that the mean determines where on the real line it's centered. So if we have a mean of zero, we know that our distribution is going to be centered around zero. And as we move further away from zero, uh, the probability is going to decrease now. Our standard deviation determines the thickness essentially of our distribution. If we have a very low standard deviation, we know that our normal distribution is going to be pretty skinny. Whereas if we have a high standard deviation, our distribution is going to be pretty wide. So knowing that our shape is completely determined by our standard deviation and our center is determined by our mean, we can answer our questions. So the question the first question is, will they have the same shape? Well, yes, they will, because the standard deviations are equal. So our first question is answered. Yes, because standard deviation one is equal to standard deviation to now. The second part of this question wants to know if they'll be centered at the same place, but we can see that they have means that are different on their first distributions. Negative four. On our second, it's positive. Six. So these are going to have different centers, which means no, they will not be centered at the same place. Negative four is not equal to six. And so these are your final answers. They will have the same shape but be centered at different places.