Question

Find these probabilities for the standard normal random variable \(z\) : a. \(P(z<2.33)\) b. \(P(z<1.645)\) c. \(P(z>1.96)\) d. \(P(-2.58

Short answer

Answer: The probabilities are a. P(z<2.33) = 0.9901, b. P(z<1.645) = 0.9500, c. P(z>1.96) = 0.0250, and d. P(-2.58<z<2.58) = 0.9900.

Step-by-step solution

Understand the given conditions for the probabilities

We are given four conditions, and for each of them, we need to find the probability of z under that condition: a. \(P(z<2.33)\) b. \(P(z<1.645)\) c. \(P(z>1.96)\) d. \(P(-2.58<z<2.58)\) To find these probabilities, we will use the Z-table.

Looking up the Z-table for each condition

For conditions (a), (b), and (d), we need the area to the left, which can be looked up directly in the Z-table. For condition (c), we will need to obtain the area to the right, which can be calculated by subtracting the area to the left (which can be found in the Z-table) from 1. a. \(P(z<2.33)\): Find the area to the left of 2.33 in the Z-table. The value is 0.9901. b. \(P(z<1.645)\): Find the area to the left of 1.645 in the Z-table. The value is 0.9500. c. \(P(z>1.96)\): Find the area to the left of 1.96 in the Z-table and then subtract it from 1 to find the area to the right of 1.96. The area to the left of 1.96 is 0.9750, so the area to the right is \(1 - 0.9750 = 0.0250\). d. \(P(-2.58<z<2.58)\): Find the area to the left of 2.58 and the area to the left of -2.58 in the Z-table, and subtract the smaller area from the larger area. The area to the left of 2.58 is 0.9950, and the area to the left of -2.58 is 0.0050. So the probability is \(0.9950 - 0.0050 = 0.9900\).

Reporting the probabilities

Now we have found the probabilities for each condition: a. \(P(z<2.33) = 0.9901\) b. \(P(z<1.645) = 0.9500\) c. \(P(z>1.96) = 0.0250\) d. \(P(-2.58<z<2.58) = 0.9900\) These are the probabilities for the standard normal random variable \(z\) under the given conditions.

Key concepts

Probability and Statistics

Probability and statistics are branches of mathematics that deal with the analysis of random events. The concept of probability quantifies how likely an event is to occur, which can range from 0 (impossible event) to 1 (certain event). Understanding probability is crucial for interpreting statistical results and making predictions about future occurrences based on past data.

Within the arena of probability, various types of distributions describe how the probabilities of outcomes are dispersed. The normal distribution is a commonly used distribution in statistics because it models many natural phenomena well. Within this context, we come across normal random variables which are associated with normal distributions.

Z-table

A Z-table, also known as the standard normal table, is a mathematical table that allows us to find the probability of a normal random variable falling within a particular range. The Z-table is derived from the standard normal distribution, which is a special case of the normal distribution with a mean (average) of 0 and a standard deviation of 1.

The table provides cumulative probabilities for different Z-value ranges. These values represent the area under the standard normal curve to the left of a given Z-value. It's a crucial tool for statisticians and researchers for calculating the probability that a statistic is observed under the standard normal distribution.

Normal Random Variable

A normal random variable refers to the variable in question when dealing with normal probability distributions. Normal distributions are symmetric, bell-shaped distributions characterized by their mean and variance.

A standard normal distribution takes a normal distribution with any mean or variance and standardizes it so the mean is 0 and the variance is 1. This process allows us to use the Z-table to find probabilities for any normal random variable by converting it into a 'Z-score'. A Z-score measures the number of standard deviations an observation or datum is above or below the mean. In the context of our exercise, z is the standard normal random variable.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a real-valued random variable is a function that indicates the probability that the variable will take a value less than or equal to a certain value. In essence, it provides the cumulative probability associated with a distribution.

For the standard normal distribution, the CDF is often represented as Φ(z), where Φ is the function name and z is the value of the random variable. It essentially tells us 'how much' of the data falls below a particular value. This function is extensively used to solve problems like the ones presented in the exercise, by finding areas under the curve of a standard normal distribution graph.