Question

Let \(z\) denote a random variable having a normal distribution with \(\mu=0\) and \(\sigma=1 .\) Determine each of the following probabilities: a. \(P(z < 0.10)\) b. \(P(z < -0.10)\) c. \(P(0.40 < z < 0.85)\) d. \(P(-0.85 < z < -0.40)\) e. \(P(-0.40 < z < 0.85)\) f. \(P(z > \- 1.25)\) g. \(P(z < -1.50\) or \(z > 2.50)\)

Short answer

Use standard normal distribution table to find the values corresponding to Z-scores. Be vigilant about the negative Z-scores, use the symmetric property of normal distribution where necessary.

Step-by-step solution

Understanding Z-Scores and Standard Normal Tables

A Z-Score is a measurement of how many standard deviations a particular data point is from the mean. It is calculated by subtracting the mean from a particular data point (in this case already given as Z-scores), and dividing this by the standard deviation. Standard Normal Tables (also known as Z-tables) is a mathematical table that allows us to know the percentage of values below (to the left of) a z-score in a standard normal distribution. Here, since we are dealing with a normal distribution where \(\mu = 0\) and \(\sigma = 1\), our Z-score is equal to the value of z itself.

Calculate Probability for Each Case

We will refer to the standard normal distribution table to find the probabilities. a. \(P(z < 0.10)\): Find the value from the standard normal table that corresponds to 0.10. That value is the probability. b. \(P(z < -0.10)\): Here, since it's a negative value, we use symmetry properties of the normal distribution which states probability of being below -0.10 is same as the probability of being above 0.10; it equals \(1 - P(z < 0.10)\). c. \(P(0.40 < z < 0.85)\): To find the probability between two values, we find the probability of lower bound and upper bound separately from the standard normal distribution table and subtract the former from the latter i.e., \(P(z < 0.85) - P(z < 0.40)\). d. \(P(-0.85 < z < -0.40)\): Similar to the previous case, we subtract the lower bound from the upper bound but as these are negative, we use the symmetry property which states probability of being below -0.85 is same as probability of being above 0.85 and likewise for -0.40. Emerges as \(P(z > 0.40) - P(z > 0.85)\). e. \(P(-0.40 < z < 0.85)\): This is calculated as \(P(z < 0.85) - P(z > 0.40)\). f. \(P(z > -1.25)\): Probability of being above -1.25 is same as probability of being below 1.25 as per symmetric property, calculated as \(P(z < 1.25)\). g. \(P(z < -1.50\) or \(z > 2.50)\): Since these are disjoint events we add the corresponding probabilities i.e., \(P(z > -1.50) + P(z < 2.50)\).

Key concepts

Z-Scores

Understanding Z-Scores is crucial when working with normal distribution probability. A Z-Score essentially tells us how far from the mean, in standard deviations, a particular score lies. It's calculated by subtracting the mean from the score and then dividing by the standard deviation. In simple terms, if you score above the mean, you have a positive Z-Score; below the mean, a negative Z-Score.

For instance, a Z-Score of 1.0 signifies that the value is one standard deviation above the mean. Similarly, a Z-Score of -1.0 indicates one standard deviation below the mean. This concept supplies a standard way to compare scores from different distributions and is especially useful because it enables us to calculate the probability of a score occurring within a normal distribution.

When you see problems involving frequencies within certain ranges, you'll often be working with Z-Scores to find these probabilities.

Standard Normal Distribution

The Standard Normal Distribution is a special case of the normal distribution that has a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. It is also referred to as the Z-Distribution. When you convert a normal distribution to a Z-Score, you are standardizing the distribution.

This means that no matter what the original mean or standard deviation was, the standard normal distribution allows us to compare scores across different scales. The shape of the standard normal distribution is the bell curve, which is symmetrical about the mean (zero in this case).

The total area under this curve equals 1, or a probability of 100%, and each segment of the curve corresponds to a proportion of that total probability. Knowing how to interpret the standard normal distribution is vital for finding probabilities related to specific Z-Scores.

Standard Normal Tables

Standard Normal Tables, often referred to as Z-Tables, are a statistical tool used to find the probability of a Z-Score occurring within the standard normal distribution. These tables are integral because they give us the area to the left (or sometimes the right) of any Z-Score on the curve, which corresponds to the cumulative probability up to that point.

To use these tables, you locate the Z-Score in question along the left column and top row of the table. The intersecting value inside the table gives you the cumulative area or probability. It's critical to note whether you're finding the area to the left (less than) or to the right (greater than) because sometimes you may need to subtract the table value from 1 to find the latter.

Understanding how to use these tables allows you to efficiently find the probabilities for different segments of the standard normal curve without the need for complicated integrations or calculations.

Probability Calculation

Probability Calculation in a standard normal distribution is about determining the likelihood of a Z-Score falling within a particular range. To calculate these probabilities, first, determine if the Z-Score is looking for a cumulative area to the left, right, or between two values.

If you have a single Z-Score, like in the problems \(P(z < 0.10)\) or \(P(z > -1.25)\), you'll use the standard normal table to find the cumulative probability up to that Z-Score. For Z-Scores that fall between two values, you'll find the cumulative probabilities for each Z-Score separately and then determine the difference between them.

It's also essential to remember the symmetry of the standard normal distribution when dealing with negative Z-Scores. This symmetry can simplify calculations, as you can often subtract the cumulative probability from 1 or find equivalencies between negative and positive Z-Scores. Mastering these calculations is fundamental in statistics, as it provides a foundation for more complex inferential statistics.