Question

Find endpoint(s) on a \(N(0,1)\) density with the given property. P.137 (a) The area to the left of the endpoint is about 0.10 (b) The area to the right of the endpoint is about 0.80 (c) The area between \(\pm z\) is about 0.95 .

Short answer

The endpoints for the given areas are approximately -1.28 for 0.10 area to the left, -0.84 for 0.80 area to the right and \(\pm 1.96\) for 0.95 area in between.

Step-by-step solution

Find the endpoint of 0.10 area to the left

The z-table will be used to find the z-score that corresponds to an area of 0.10 to its left. Reading the z-table, the closest value to 0.10 is 0.1003 corresponding to a z-score of -1.28. Therefore, the endpoint on a \(N(0,1)\) density such that the area to the left of the endpoint is about 0.10 is approximately -1.28.

Find the endpoint of 0.80 area to the right

The z-table shows the probability to the left of a given z-score. Therefore, to find a z-score with 0.80 probability to its RIGHT, you'll find a z-score with 0.20 (1-0.80) probability to its LEFT. The closest value to 0.20 on the z-table is 0.2005, corresponding to a z-score of -0.84. The endpoint on a \(N(0,1)\) density such that the area to the right of the endpoint is about 0.80 is approximately -0.84.

Find the endpoints of 0.95 area between \(\pm z\)

Here, the request is about two endpoints, which divide the standard normal distribution such that there is 0.95 probability between them and 0.025(=0.5*(1-0.95)) to either side of them. Looking at the z-table, the values closest to 0.025 at the tails are 0.0250 and 0.0256, both corresponding to z-scores of \(\pm 1.96\). Thus, the endpoints on a \(N(0,1)\) density such that the area between them is about 0.95 are approximately \(\pm 1.96\).

Key concepts

Standard Normal Distribution

The standard normal distribution, denoted as \( N(0,1) \), is a critical concept in statistics characterized by a mean of zero \( (\mu = 0) \) and a standard deviation of one \( (\sigma = 1) \). This distribution is foundational in probability and statistics because it serves as a reference point from which all other normal distributions can be derived through standardization (z-score transformation).

In the context of the given exercise where you're asked to find specific endpoints, the standard normal distribution allows you to use standardized tables (z-tables) to locate the z-scores corresponding to different areas under the curve. For instance, when you're looking for the z-score associated with an area of 0.10 to the left, the symmetry of the standard normal distribution simplifies this task, as it's simply the negative of the z-score for the same area to the right of the mean.

The z-score itself represents how many standard deviations an element is from the mean. When a distribution is standard normal, this z-score can directly tell you the probability (area under the curve) of randomly drawing a certain value from the distribution.

Probability

Probability is the measure of the likelihood that an event will occur. It ranges from 0 (impossibility) to 1 (certainty). In a standard normal distribution, the probability of an event is visualized as the area under the curve corresponding to a range of values.

For example, locating a z-score for a given probability involves finding the equivalent area under the curve of the standard normal distribution. As shown in the exercise's step-by-step solution, to find an endpoint with a 0.80 probability to its right, you calculate the complementary probability to the left, which is 0.20. By consulting the z-table, you identify the z-score that corresponds to an area (or probability) of 0.20 to the left of the z-score. Understanding the relationship between area under the curve and probability allows you to navigate the z-table which tabulates these values in a way that facilitates these calculations in the context of the normal distribution.

Area Under the Curve

The area under the curve in a standard normal distribution graph represents cumulative probability. This area is essential in determining the likelihood of a random variable falling within a particular range. The total area under the curve for any probability density function, including the standard normal distribution, is always equal to 1, or 100% probability.

When solving exercises like those provided, which involve finding endpoints based on areas (probabilities), the symmetric property of the standard normal curve is a valuable tool. For instance, if the area to the left of a z-score is 0.10, the corresponding area to the right is 0.90, because the total area must add up to 1. Similarly, for a z-score that creates a central area of 0.95, you can deduce that there is 0.025 on each side of the distribution's tails. Understanding how to interpret and manipulate the area under the curve of a standard normal distribution is integral to solving many problems in statistics, such as calculating confidence intervals or setting thresholds for hypothesis testing.