Question

a. Find a \(z_{0}\) such that \(P\left(-z_{0} \leq z \leq z_{0}\right)=.95\). b. Find a \(z_{0}\) such that \(P\left(-z_{0} \leq z \leq z_{0}\right)=.98\).

Short answer

Question: Determine the values of \(z_0\) such that the probability density function for the standard normal distribution results in the following probabilities: a. \(P\left(-z_{0} \leq z \leq z_{0}\right)=.95\) and b. \(P\left(-z_{0} \leq z \leq z_{0}\right)=.98\). Answer: The values of \(z_0\) are approximately 1.96 for a 95% probability and 2.33 for a 98% probability.

Step-by-step solution

Part a: Find \(z_0\) for a 95% probability

To find the \(z_0\) value for a 95% probability, we need to locate the \(z\) value in the standard normal distribution table associated with the probability of 0.475 (since the total probability is 0.95 and the region is symmetric about the mean). We find that the \(z\) value corresponding to this probability is approximately 1.96. Therefore, we have: \(z_0 \approx 1.96\)

Part b: Find \(z_0\) for a 98% probability

Similarly, for a 98% probability, we need to find the \(z\) value in the standard normal distribution table associated with the probability of 0.49. We find that the \(z\) value corresponding to this probability is approximately 2.33. Therefore, we have: \(z_0 \approx 2.33\) In summary, the values of \(z_0\) for the given probabilities are approximately 1.96 for a 95% probability and 2.33 for a 98% probability.

Key concepts

Z-scores

Z-scores are a fundamental concept when working with standard normal distributions. At its core, a Z-score represents how many standard deviations away a data point is from the mean. Z-scores allow us to understand the position of a particular value within the distribution.

In a standard normal distribution, the mean is 0, and the standard deviation is 1. This makes it easy to calculate Z-scores. If you have a data point, you subtract the mean from it, then divide by the standard deviation. The result is the Z-score.

• Z-scores help in comparing different data points from various normal distributions.
• Positive Z-scores indicate values above the mean, while negative Z-scores indicate values below.
• A Z-score of 0 implies the data point is exactly at the mean.

Understanding Z-scores is crucial for interpreting data within a normal distribution, and it forms the basis for using probability tables to find probabilities associated with Z-scores.

Probability Tables

Probability tables, often called Z-tables, are tools used to find the probability of a Z-score occurring below, above, or between certain values in a standard normal distribution. These tables consolidate the cumulative probabilities associated with Z-scores so one can easily find the probability linked to a specific Z-score.

To use a Z-table, first determine your Z-score. Then, find this Z-score in the table to obtain the probability. The probability gives the area under the curve to the left of your Z-score in a standard normal distribution.

• Probability tables typically provide probabilities for the left tail of the distribution. You may need to subtract from 1 for right tail probabilities.
• For central probabilities, like in the exercise, find the probability for the Z-score and double it, considering the symmetry of the distribution.
• They help transform raw data into meaningful probability information, allowing us to make statistical decisions.

Using these tables reduces the need for complex calculations and provides a quick way to grasp the likelihood of outcomes within standard normal distributions.

Symmetric Distribution

A symmetric distribution is one where the left and right sides are mirror images of each other. In the context of a normal distribution, which is the case for the standard normal distribution, symmetric refers to the perfect bell curve shape.

This symmetry is incredibly useful in probability and statistics. It means that the probabilities on either side of the mean can be treated equally. For example, a 95% confidence interval around the mean splits into two equal tails of 2.5% each.

• This symmetry simplifies calculations and interpretations of probabilities and high-impact statistical metrics.
• When solving parts of the exercise, the symmetry is used to locate corresponding Z-scores for given probabilities like 95% or 98% around the mean.
• It confirms that for any given Z-score, the negative equivalent has the same probability magnitude.

Understanding symmetric distribution is pivotal for recognizing how probabilities distribute in a standard normal distribution and allows seamless application of Z-scores and probability tables.