Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.


Less than -2.00

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves a standard normal distribution, which has a mean (μ) of 0 and a standard deviation (σ) of 1. You are tasked with finding the probability that a randomly selected test score is less than -2.00.
2. Step 2: Visualize the problem. Draw a standard normal distribution curve (bell-shaped curve) with the mean at 0. Mark the value -2.00 on the horizontal axis, which is to the left of the mean. Shade the area under the curve to the left of -2.00, as this represents the probability we are trying to find.
3. Step 3: Use the standard normal distribution table (Z-table) or technology. The Z-table provides cumulative probabilities for Z-scores (standardized values). Locate the Z-score of -2.00 in the table to find the cumulative probability, which represents the area under the curve to the left of -2.00.
4. Step 4: If using technology (e.g., a calculator or statistical software), use the cumulative distribution function (CDF) for the standard normal distribution. Input the Z-score of -2.00 to calculate the cumulative probability. For example, in a calculator, you might use a function like P(Z < -2.00).
5. Step 5: Interpret the result. The cumulative probability you find represents the likelihood that a randomly selected test score is less than -2.00. Ensure the result is rounded to four decimal places, as specified in the problem.

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the Z-score, which indicates how many standard deviations an element is from the mean. This distribution is symmetric and bell-shaped, making it useful for calculating probabilities and percentiles for normally distributed data.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores allow for the comparison of scores from different normal distributions by standardizing them, making it easier to find probabilities using the standard normal distribution.

Probability and Area Under the Curve

In the context of the normal distribution, probability is represented by the area under the curve of the distribution graph. To find the probability of a score being less than a certain value, one can calculate the area to the left of that value on the standard normal distribution curve. This area can be found using Z-tables or technology, providing insights into how likely a score falls within a specific range.