1. Use a normal distribution calculator or Excel to find the...

1. Use a normal distribution calculator or Excel to find the following probabilities for a standard normal random variable Z~Normal(? = 0, ? = 1). If you're using Excel, draw a picture of the normal distribution - is your answer reasonable? P(0 < Z < 0.50) 0.6915 - 0.5 = 0.1915 P(2 < Z < 3) P(-0.50 < Z < 0.50) P(Z < -1.96) P(Z > 0) P(Z > 1.96) P(Z = 0) P(Z < 1.65) P(1.22 < Z < 2.15) P(Z > -1.65) P(Z > 1.22) P(-1.96 < Z < 1.96)

Transcript

00:01 Okay, so in this problem, we are dealing with the standard normal distribution, which is a normal distribution that has a mean of zero, because we're talking about z scores, and the standard deviation is one.

00:15 So you can recall from your empirical rule that 68 % will be within one standard deviation of the mean, 95 % within two standard deviations, and then 99 .7 within three standard.

00:35 Deviations of the mean.

00:37 So the first one we're going to do here is normal cdf from negative 0 .5 to positive 0 .5.

00:45 So what that's going to look like is basically this area right here in the middle with a mean of zero and standard deviation of 1.

00:53 So negative 0 .5 to positive 0 .5 mean and standard deviation is 0 .32 or 383 is what it rounds to actually if we're doing three decimals rounding the next one is z greater than zero that one's really easy we don't even have to worry about the calculator because that is exactly half of the half of the graph which is going to just be 0 .5 equal zero that is basically if it's equal zero that's like a normal pdf kind of problem the the probability of getting a z score of exactly zero.

01:37 Since this is a continuous distribution, that is zero.

01:42 I can get any value basically between negative three and positive three, any real number.

01:47 So the probability of getting exactly 0 .000 is really impossible.

01:55 All right, so normal cdf from 1 .22 to 2 .15, 0 and 1.

02:03 And at this point it's kind of repetitive we're doing essentially the same thing using that normal curve calculator and you can kind of shade it and make sure it makes sense this one is a 0 .095 just greater than 1 .22 so that's going to go up to infinity you can put in something to represent that and again just the normal cdf function there 1 .22 up to some representation of infinity is 0 .11 another one there at the end.

02:50 Moving to the second column, normal cdf between 2 and 3 with a mean of 0 and standard deviation of 1...

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