A variable is normally distributed with mean 0 and standard...

A variable is normally distributed with mean 0 and standard deviation 4 a. Determine and interpret the quartiles of the variable. b. Obtain and interpret the second decile. c. Find the value that $15 \%$ of all possible values of the variable excecd. d. Find the two values that divide the area under the corresponding normal curve into a middle area of 0.80 and two outside areas of 0.10. Interpret your answer.

A variable is normally distributed with mean 0 and standard deviation 4
a. Determine and interpret the quartiles of the variable.
b. Obtain and interpret the second decile.
c. Find the value that $15 \%$ of all possible values of the variable excecd.
d. Find the two values that divide the area under the corresponding normal curve into a middle area of 0.80 and two outside areas of 0.10. Interpret your answer.

Key Concepts

Percentiles

Percentiles are measures indicating the value below which a given percentage of observations in a dataset fall. In the context of the normal distribution, percentiles allow for the determination of cutoff points for the data, and are often used to identify extreme values or threshold points, such as the value below which 15% of all possible values lie.

Standardization

Standardization is the process of converting a normal variable with any given mean and standard deviation into a standard normal variable with mean zero and standard deviation one, typically using the z-score formula. This transformation facilitates the lookup of probabilities and critical values using standard normal distribution tables, thereby simplifying the process of finding specific quantiles like quartiles, deciles, or percentiles.

Deciles

Deciles divide a data set into ten equal parts, with each decile representing 10% of the data. The second decile, for example, corresponds to the 20th percentile of the distribution. Deciles are useful in providing a more detailed quantile breakdown of the data, especially when assessing the distribution of values.

Quartiles

Quartiles are values that divide a data set into four equal parts. In a normal distribution, quartiles are determined by finding the points below which certain percentages (25%, 50%, and 75%) of the data lie. These measures help in understanding the spread and dispersion of the data around the median.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is fully characterized by its mean and standard deviation, which determine the center and spread of the distribution respectively.

Transcript

00:01 So we're looking at a normally distributed mean and standard deviation of four.

00:06 We're trying to find the quartiles of the variable.

00:10 So the first way we're going to do that is that there are three quartiles, even though quart sounds like four.

00:16 They are q1, q2, and q3, where q2 is the median.

00:21 Just a fun fact there.

00:23 And what we're looking for here is the z score in the back of the book.

00:28 So if we split up our graph, we'll notice that down the middle.

00:35 I can't go out that high.

00:37 But this is where 50 % of the data is.

00:42 You've got 50 % of the data of the left and 50 % of the data the right.

00:47 And now 25 % of the data is going to be over here.

00:50 Not necessarily in that extreme, but maybe it's moved over a little bit.

00:54 I just split up this line equally, not necessarily in relation to the graph.

01:00 But we're going to look for 25%, and this is going to be at the z score of 0 .25.

01:11 Or no, the z scores corresponding to 25 % of this graph.

01:17 And what we find is that this first one is 67%, or sorry, a z score of negative.

01:26 0 .67.

01:28 The one in the middle is 0.

01:31 And this one of the right, you might have guessed it, is 0 .67.

01:37 So the corresponding values for the quartiles is going to be as follows.

01:44 So for x1, we are given 0 minus 0 .67.

01:52 This could be a plus, but i went ahead and skip that step, because we're multiplying, adding a, we're adding a negative, which is the same as subtracting.

02:02 Then we'll multiply here by 4, since that's our standard deviation.

02:08 And what we get is negative 2 .68.

02:13 So 25 % of all observations will be smaller than this.

02:19 Now for x2, that's just going to be 0 since we're doing 0 minus something times 0.

02:26 So 50 % of all observations are smaller than zero.

02:33 Now for the 75 % of all observations, that is going to be 0 plus 2 thirds of 4.

02:42 And that is going to be 2 .68, unsurprisingly.

02:51 All right.

02:52 So the next part, we are asked to find the value that 15 % of all possible values of the variable exceed...