Find the z value to the right of the mean so that a. 54.78% of the area under the distribution c

Find the z value to the right of the mean so that a. 54.78%...

Key Concepts

Percentiles

Percentiles indicate the relative standing of a value within a dataset by showing the percentage of data points that are below it. In standard normal distribution problems, specific percentiles correspond to particular z-scores, which are determined using the inverse of the CDF.

Standard Normal Distribution

This is a special normal distribution with a mean of 0 and a standard deviation of 1. It serves as a foundational model in statistics for comparing different datasets and for the evaluation of probabilities and percentiles, due to its symmetry and well-defined properties.

Z-Scores

A z-score represents the number of standard deviations a data point is from the mean of the distribution. It is used to determine the relative position of a value within the standard normal distribution and is pivotal for finding corresponding probability values or percentiles.

Cumulative Distribution Function (CDF)

The CDF of a distribution gives the probability that a random variable will take a value less than or equal to a specific value. In the context of the standard normal distribution, the CDF is used to find the area under the curve to the left of a given z-score, representing cumulative probability.

Transcript

00:01 Okay, in order to answer this question, let's remind ourselves what a z value is or what a z score is.

00:09 So if i'm looking here, i'm going to go ahead and sketch a normal curve or what's close to a normal curve.

00:16 I'm going to draw a dog line right here down the middle where the mean would be.

00:20 So if i have a set of data or a set of values that can be modeled using a normal distribution, a z score tells me how, many standard deviations a value is away from the mean of that data set.

00:37 So if i have a z score of 1, that means that value is going to be one standard deviation away from the mean.

00:43 A z score of 2 would mean i have a value that's two standard deviations away from the mean.

00:49 Negative 1 would be one standard deviation below the mean.

00:53 Negative 2 would be 1, 2 standard deviations below the mean, etc, et cetera, et cetera.

00:59 So let me kind of erase this here and redraw my normal curve.

01:05 So notice when i draw my normal curve here, when i draw my normal distribution, have this curve here.

01:13 And i'm going to go ahead and draw this dotted line to represent the mean.

01:16 Notice that 50 % or half of my z scores or my z values are going to be lower than this point right here.

01:26 This mean point right here, the z score right there represents zero because i'm zero standard deviations away from the mean.

01:34 50 % of my z scores are going to be less than that.

01:38 So the probability, let me write it this way.

01:42 You don't have to write it this way, but the probability of having a z score less than zero, probability of having a z score less than zero is going to be 50%.

01:56 The probability of having a z score less than 0 is 50%.

01:59 Okay.

02:00 So what we're asked to find is we're trying to find the z score where 54 .78 % of the area under the curve is to the left of that z score.

02:16 So this is part, the first part of our problem.

02:20 So if you look at my normal curve, 50 % of my z scores or 50 % of the area under this curve, is going to be left of zero, 54 .78 % of my z scores are going to be less than maybe this point right here, ish.

02:37 Okay, so 54 .78 % of my z scores or 54 .78 % of the area under this curve is going to be to the left of this z value right here.

02:47 Well, conveniently, we have a way of finding what that z value is based on this percentage.

02:54 So either in your textbook or if you are to look anywhere on the internet, just search like z score table.

03:00 You'll pop up with a bunch of different z score tables.

03:03 A z score table is basically a table with a bunch of z values, a bunch of z scores, and the corresponding areas under the curve to the left of those z scores.

03:18 So here i have this percentage, 54 .78%.

03:23 Or i'm going to write that as a decimal 0 .5478.

03:26 So if i go to my z table, and again, you can pull up a z table in your textbook or on the internet, i'm going to go to my z table and find in the meat and potatoes of my z table, i'm going to find the percentage or the value 0 .5478.

03:42 And looking at my table, conveniently, i can see a value exactly that's 0 .5478.

03:49 If i didn't have a 0 .5478 on there, i'd find the value that was closest...

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