Problem 34 - An Introduction to Mathematical Statistics and Its Applications

Question

Assume that the random variable \(Z\) is described by a standard normal curve \(f_{Z}(z)\). For what values of \(z\) are the following statements true? (a) \(P(Z \leq z)=0.33\) (b) \(P(Z \geq z)=0.2236\) (c) \(P(-1.00 \leq Z \leq z)=0.5004\) (d) \(P(-z

Step-by-Step Solution

Answer

(a) -0.44, (b) 0.77, (c) 0, (d) ±1.28, (e) 0.94

Step 1: Find z-value for P(Z ≤ z) = 0.33

Use your Z-table or calculator to find the 'z' value corresponding to a left-tail probability of 0.33, which is -0.44.

Step 2: Find z-value for P(Z ≥ z) = 0.2236

This is a right-tail probability. Using your Z-table or calculator, find the 'z' value corresponding to 1-0.2236 (0.7764), which is the left-tail probability. The corresponding 'z' value is 0.77.

Step 3: Find z-value for P(-1.00 ≤ Z ≤ z) = 0.5004

We know -1.00 already encompasses about 0.3413 (from Z-table or calculator. 0.5004-0.3413 = 0.1591 is now the remaining lower-tail probability to find. Find the corresponding 'z' value from Z-table or calculator for 0.3413+0.1591 = 0.5004, which is 0.

Step 4: Find z-value for P(-z<Z<z) = 0.80

This is the probability inside the two tails from -z to z. Since the normal distribution is symmetrical, each tail contains half of the probability of 0.20 (1-0.80). So find the 'z' value corresponding to an upper (or lower) tail of probability 0.10. The corresponding 'z' value found from Z-table or calculator is approximately ±1.28.

Step 5: Find z-value for P(z ≤ Z ≤ 2.03) = 0.15

We know from the Z-table or calculator that Z=2.03 corresponds to about 0.9792 cumulative probability. We are looking for the 'z' which encompasses the remaining probability. So, 0.9792-0.15=0.8292 is the left-tail probability to find. The corresponding 'z' value from Z-table or calculator is approximately 0.94.

Key Concepts

Z-table

The Z-table, also known as the standard normal table or unit normal table, is a mathematical table that allows us to determine the percentage of values that lie below a certain Z-score in a standard normal distribution. A Z-score, or standard score, represents how many standard deviations an element is from the mean.

To use a Z-table, you first need to calculate the Z-score for the given value, then find this score on the table to see the percentage (or probability) below that Z-score. Z-tables come in different formats, some show the cumulative probability from the left up to a z-value, while others provide the cumulative probability from the right. In solving mathematical problems, it's crucial to know which part of the table you are looking at, as this will determine how you interpret and use the values to find probabilities.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is graphically represented by a bell-shaped curve, known as the standard normal curve.

One of the defining characteristics of the standard normal distribution is that it has a mean of 0 and a standard deviation of 1. This makes it easy to compare different data sets by converting their values to Z-scores. Key properties of the normal distribution include its symmetry, where the left half of the distribution mirrors the right half, and its tails that approach, but never touch, the horizontal axis.

Cumulative Probability

Cumulative probability is the likelihood that a random variable will take a value less than or equal to a certain number. In the context of standard normal distribution, it represents the area under the curve to the left of a Z-score.

When you're given a normal distribution problem, determining the cumulative probability can help you understand what portion of data falls below a certain threshold. By using cumulative probability, you can also answer broader questions about the data set, such as percentile rankings or the probability of a value falling within a specific range.

Z-value Calculation

The calculation of a Z-value, or Z-score, is essential in figuring out where a particular data point lies in relation to the mean of a distribution. The Z-score is calculated using the formula: \( Z = \frac{{X - \mu}}{{\sigma}} \) where \(X\) is the value in question, \(\mu\) is the mean of the data set, and \(\sigma\) is the standard deviation. In the case of the standard normal distribution, since the mean is 0 and the standard deviation is 1, the Z-score equals the value itself.

To find a Z-value from a cumulative probability, you may need to use inverse operations or refer to a Z-table or software/calculator that provides this functionality. Understanding how to calculate and interpret Z-scores is crucial for numerous applications in statistics, including hypothesis testing and confidence interval estimation.