As shown in Example 6.12, incomes of junior executives are normally distributed with a standard

As shown in Example 6.12, incomes of junior executives are...

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) measures the probability that a random variable from a given distribution will take a value less than or equal to a specified threshold. In the context of the normal distribution, the CDF helps to determine the likelihood of a variable falling below a certain value and is fundamental when computing probabilities based on percentiles.

Z-Score Transformation

The Z-score transformation converts a data value from a normal distribution into a standard score, which represents the number of standard deviations that value is from the mean. This standardization process is critical as it simplifies comparisons across different normal distributions and facilitates the use of standard normal probability tables.

Percentiles

Percentiles are values that divide a data set into 100 equal parts, showing the relative standing of a value within the distribution. For example, being at the top end of the middle 80% implies a specific percentile cutoff that can be used to relate specific values to their probability within a normal distribution.

Standard Deviation

Standard deviation is a measure of dispersion within a set of data values. In the context of a normal distribution, it describes the spread of the data around the mean. A smaller standard deviation indicates that the data points are clustered closely around the mean, while a larger standard deviation suggests that they are spread out over a wider range.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean. It is characterized by a bell-shaped curve where most observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions. This distribution is foundational in statistics for modeling real-world phenomena, particularly those that tend to cluster near the mean.

Transcript

00:02 Okay, for this problem, we have some more normally distributed data, but this problem is a little bit different because we don't know what the means for is.

00:12 So with the most normal curve problems, we want to sketch the normal distribution and write down what we know.

00:19 So the first thing you want to do is write down we know that standard deviation equals to $3 ,898.

00:32 So for junior executives at a company.

00:36 So what's different about this problem than some of the earlier problems is that we want for this problem is we want to know for part a at least.

00:46 We want to know what the mean value is.

00:50 So our plan is going to be to really use the formula for the z score to get the mean.

01:01 So we're going to have to find the z score.

01:03 So let's figure out how to get a z score here.

01:06 So first things first, we're going to get a, go to our case.

01:09 Calculator and we're told in the problem that there's an 80 % the middle 80 % of the incomes have it a certain we have the high end of the salary for the 80 % of the incomes we know what that is so let's do something here we call the inverse normal we've been doing forward what i call forward problems we put in the the z scores that we know are put in the parameters that we know and to get the probabilities this case we know the amount of probability of the area is 80%.

01:41 So that's 80.

01:47 And so we're going to work backwards to get the z score that goes with 80%.

01:52 And figure out what that is.

01:53 So i'm going to leave the mean at zero and the standard deviation at 1, because we're going to go a z score.

02:02 And let me just mark this up here too.

02:06 And so the number that we know is the high end of the 80%.

02:12 So that's the 0 .80 number.

02:15 Kind of from there back is 62 ,900.

02:23 So we know that case, 62 ,900 is the high end of the middle 80 % of the earners.

02:29 We want to know is the mean here.

02:33 So from our inverse normal calculation, we work backwards on the c -t8mull technically.

02:39 So we know that we can use a z score.

02:41 So let's go back to our plan here.

02:43 So now we know the z score that goes with the number that we have.

02:48 The 62 ,900 is .84.

02:52 So that's the z score that goes with it.

02:55 And so we would know using the z that 62 ,900 minus the mean divided by the standard deviation, which we stated over there is 3898 is the relationship.

03:11 So we can work backwards to find out what that mean number is for this overall distribution.

03:16 And then part b, we're going to do a little bit more work to figure out a different parameter.

03:22 So i'm going to use my calculator since it's up here.

03:24 So i see 0 .84.

03:26 I'm going to solve for the mean values...

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