Find two z values, one positive and one negative, that are equidistant from the mean so that the

Find two z values, one positive and one negative, that are...

Key Concepts

Inverse Cumulative Distribution Function (Quantile Function)

The inverse cumulative distribution function, also known as the quantile function, is used to determine the z-score corresponding to a given cumulative probability. In this context, it allows you to find the specific z-values where the cumulative probability up to that point equals a preset value derived from the tail area specifications. This function is fundamental in statistical methods for finding critical values in both one-tailed and two-tailed tests.

Standard Normal Distribution

The Standard Normal Distribution is a continuous probability distribution with a mean of 0 and a standard deviation of 1. It is symmetric about the mean, which implies that z-scores equidistant from the mean correspond to equal probabilities in the left and right tails. This property is essential when looking for z-values that contain specific tail areas.

Tail Areas

Tail areas in a normal distribution refer to the regions in the extremes of the distribution beyond a certain z-value. When the problem specifies that the areas in the two tails add to a given percentage, it implies that each tail has half of that percentage, due to symmetry. Understanding tail areas is crucial for determining cutoff values in hypothesis testing and confidence interval estimation.

Transcript

00:01 Okay, this problem is finding 2 z values, 1 positive, 1 negative, that are equal distance from the mean of normal distribution, so that the areas in the two tails add up to be 5%, 10 % and 1%.

00:15 Okay, so question 1, the area of 2 tails add up to be 5%.

00:20 Here is the mean of 0, which is 0, mean of the normal distribution, which is 0.

00:25 And we have two tails, one tail and the other tail, and there have 2 z value, z1 and z2.

00:34 And they are equal distance from the mean zero.

00:37 Therefore, z2 will be negative z1.

00:41 And the two tails, according to the question, add up to be 5%, which is 0 .05.

00:51 Therefore, each of the tail would count half of the value 5%.

00:56 So we have 0 .05 divided by 2, which is 0 .025.

01:02 Then based on this value, we can calculate the z1.

01:07 So z1 would be obtained from inverse normal distribution.

01:19 We can do that in decimals.

01:28 Okay, here we go.