(a) For the normal distribution shown in Figure 1, find...

(a) For the normal distribution shown in Figure 1, find approximate values for its mean and standard deviation. Explain how you obtained your answers. -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Figure 1 (b) The normal distribution of a variable x has mean ? = 110 and standard deviation ? = 20. (i) Sketch this distribution. Explain and show any calculations used to determine the key features of the distribution. (ii) Write down the formula for z that converts each value of the variable x so that z follows a standard normal distribution. Your answer should contain both the general formula and the specific formula for this example. (iii) Calculate the value of z corresponding to x = 80. (iv) Interpret the value of z by completing the sentence: The value of x = 80 is ______ standard deviations ______ its mean. (v) Suppose you are obtaining a sample of size 100 from this distribution. What will be the standard deviation of the sampling distribution of the mean?

Transcript

00:01 So we have this kind of normal distribution.

00:03 The middle here is in minus 3.

00:05 And we have that the curve here ends in almost like minus 9.

00:11 We have some other values here minus 10.

00:15 And here the same thing like 9.

00:17 Oh, sorry, 3 and 4, for example.

00:24 So here, for example, because this is the middle, where the distribution is symmetric around, this means that the mean of this distribution is this point, the pike.

00:37 Now to find what is the standard deviation, we need to know where almost all the observations are here within the mean, for example.

00:49 So we have the almost all the observations are within minus 9 and 3.

00:56 So considering this, like this means that if you consider what is this distance here, usually if you have a normal distribution, this will be around three standard deviations from the mean, like minus three standard deviations.

01:11 And here will be three standard deviations from the mean.

01:16 So in this case here, what we could consider is that here what we have is that the sigma, if you want to consider is, for example, if this value, if this point here is exactly three standard deviations plus the mean, which is minus 3, is equal to this value.

01:41 This means that if you want to consider here that you have exactly this, so 6 here, and then you divide this by 3, so sigma is equal to 2.

01:53 So basically what i'm using here is what we call the empirical rule, and basically the 99 .7 % of the imperial.

02:04 Which says that 99 .7 % of the observations will be within three standard deviations from the mean of the distribution.

02:12 So if i consider that this point is above three synodivations from the mean, i can find what is this value of sigma using this formula, for example.

02:23 So using this, i have the sigma here is two.

02:26 Now for the other part of this question, let's say that we have a normal distribution with mean equals to 110 and a standard deviation equals to 20.

02:37 So first we can draw this distribution so again the middle here the pike will be the mean and we need to have like in this case using the imperial rule at least that the observations almost all the observations will be within two standard deviation three standard deviations from the mean which means that like this point here could be considered as 50 and the above point will be considered 170...

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