Question

Calculate the area under the standard normal curve between these values: a. \(z=-2.0\) and \(z=2.0\) b. \(z=-2.3\) and -1.5

Short answer

Answer: The area under the standard normal curve between z = -2.0 and z = 2.0 is approximately 0.95450, and the area between z = -2.3 and z = -1.5 is approximately 0.05609.

Step-by-step solution

Part a - Find the area under the curve between z = -2.0 and z = 2.0

1. Locate the cumulative probabilities of each z value from the standard normal table or the calculator. For \(z=-2.0\), the cumulative probability is \(0.02275\). For \(z=2.0\), the cumulative probability is \(0.97725\). 2. Subtract the probabilities to find the area between the two z values. Area = \(0.97725 - 0.02275 = 0.95450\) The area under the standard normal curve between \(z=-2.0\) and \(z=2.0\) is approximately \(0.95450\).

Part b - Find the area under the curve between z = -2.3 and z = -1.5

1. Locate the cumulative probabilities of each z value from the standard normal table or the calculator. For \(z=-2.3\), the cumulative probability is \(0.01072\). For \(z=-1.5\), the cumulative probability is \(0.06681\). 2. Subtract the probabilities to find the area between the two z values. Area = \(0.06681 - 0.01072 = 0.05609\) The area under the standard normal curve between \(z=-2.3\) and \(z=-1.5\) is approximately \(0.05609\)

Key concepts

Z-Score Calculation

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point's score is identical to the mean score. To calculate the z-score of a particular value, you use the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where \(X\) is the value for which the z-score is to be calculated, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. This can be particularly useful when trying to understand how far and in what direction a value lies from the mean of the distribution, allowing comparisons between different data sets.

Cumulative Probability

Cumulative probability refers to the likelihood that a randomly chosen value from a dataset is less than or equal to a certain value. It is essentially the probability that a variable will fall within a specified range.
For the standard normal distribution, which is symmetrical with a mean of 0 and a standard deviation of 1, cumulative probabilities are tabulated for z-scores ranging from the mean to the tail ends of the curve. Calculating the cumulative probability for any z-score can provide insight into the proportion of the data that lies below (or above) that z-score. This can help in understanding various real-life probabilities, such as test scores, quality control metrics, and many other fields that rely on statistical analysis.

Area Under Normal Distribution

In a normal distribution, the total area under the curve corresponds to the probability of all possible outcomes and is always equal to 1, or 100%. The central feature of the normal distribution is that it is symmetric around the mean, which implies that the probability of observations falling within a certain number of standard deviations from the mean can be calculated.
The normal distribution model is a valuable tool because many natural and human-made phenomena are shaped by it. In the context of the exercise, calculating the area between two z-scores involves finding the cumulative probability for each z-score and then taking the difference to get the probability for the values lying in that range. This area can inform us about the percentage of observations that fall between any two given points on the distribution curve.

Standard Normal Table

A standard normal table, also known as a Z-table, is an essential tool in statistics for finding cumulative probabilities associated with the standard normal distribution. The table shows the probability that a standard normal variable is less than or equal to a given value. It typically lists z-scores to the hundredths place with corresponding cumulative probabilities.
Given that the standard normal distribution is symmetric, most tables only provide values for positive z-scores. To find the cumulative probability for a negative z-score, you would look up the positive value and subtract it from 1. The standard normal table is widely used for so many statistical applications, including hypothesis testing and setting confidence intervals, making it invaluable to students and professionals alike. When we refer to the area under the curve in exercises like the given problem, the Z-table helps us to locate these areas promptly and accurately.