Question

Find the specified areas for a \(N(0,1)\) density. P.133 (a) The area below \(z=1.04\) (b) The area above \(z=-1.5\) (c) The area between \(z=1\) and \(z=2\)

Short answer

After consulting a standard normal distribution Z-table: (a) the area below z=1.04 is approximately 0.8508, (b) the area above z=-1.5 is approximately 0.9332, and (c) the area between z=1 and z=2 is approximately 0.1359.

Step-by-step solution

Find the area below z=1.04

Consult the standard normal distribution table for z-value of 1.04. The table will directly provide the accumulated area below this z-score.

Find the area above z=-1.5

Since it's known that the total area under the curve of a standard normal distribution equals to 1, the area above a certain z-score can be calculated by subtracting the accumulated area below that z-score from 1. Thus, find the area for z=-1.5 in the standard normal distribution table and subtract this value from 1.

Find the area between z=1 and z=2

To find the area between two different z-scores, one needs to subtract the accumulated area of the lower z-score from the accumulated area of the higher z-score. Therefore, find the areas for both z-scores in table, then subtract the area found for z=1 from the one found for z=2.

Key concepts

Understanding Z-Score

The concept of the z-score is fundamental when studying the standard normal distribution, a type of normal distribution with a mean of 0 and a standard deviation of 1. A z-score, often referred to as a standard score, measures how many standard deviations an element is from the mean of its set. To compute a z-score, you subtract the mean from the data point and then divide by the standard deviation.

For instance, in the exercise, one part asked for the area below z=1.04. This indicates that we want to know the probability of a randomly chosen value from our dataset being less than 1.04 standard deviations above the mean. The z-score is particularly helpful because it allows for the comparison between different sets of data, which might have their own unique means and standard deviations, on a standard scale.

Navigating the Normal Distribution Table

A normal distribution table, also known as a z-table, is an essential tool for working with the standard normal distribution. It lists z-scores along with their corresponding percentages that reflect the area under the curve to the left of the given z-score. These tables are used to find probabilities and areas under the normal curve without manual calculation.

When the exercise asks to find the area above z=-1.5, you'd look up the absolute value of z (1.5) in the table to find the area to the left of this z-score. Since the table gives the area to the left, for z-scores below the mean (which are negative), you'd subtract this value from 1 to get the area to the right.

Area Under the Curve

To understand the area under the curve, imagine the standard normal distribution as a smooth, bell-shaped graph. This graph has a particular property where the total area under the curve adds up to 1, representing the total probability. Each point on the x-axis represents a z-score and the corresponding y-value represents the probability density at that z-score.

h4. Using Areas to Find Probabilities
To resolve the last part of the exercise, finding the area between z=1 and z=2, you subtract the accumulated area of the lower z-score from the accumulated area of the higher z-score. By doing so, you’re effectively calculating the probability of a value falling between those two standard deviations above the mean. This ability to find areas between values is one of the most powerful aspects of probability theory, relying on the properties of the standard normal distribution.