Problem 18 - Basic Statistics for Business and Economics

Question

A normal population has a mean of 80.0 and a standard deviation of \(14.0 .\) a. Compute the probability of a value between 75.0 and 90.0 . b. Compute the probability of a value 75.0 or less. c. Compute the probability of a value between 55.0 and 70.0 .

Step-by-Step Solution

Answer

a) 0.4017, b) 0.3594, c) 0.2022

Step 1: Understand the Problem

We are tasked with finding probabilities for different value ranges in a normal distribution with a mean (\(\mu\)) of 80.0 and a standard deviation (\(\sigma\)) of 14.0. We'll convert these values into z-scores and use the standard normal distribution table to find the corresponding probabilities.

Step 2: Convert to Z-scores for 75.0 and 90.0

The z-score formula is \( z = \frac{(X - \mu)}{\sigma} \). For 75.0, \( z = \frac{(75.0 - 80.0)}{14.0} = -\frac{5}{14} \approx -0.36 \). For 90.0, \( z = \frac{(90.0 - 80.0)}{14.0} = \frac{10}{14} \approx 0.71 \).

Step 3: Find the Probability for 75.0 to 90.0

Using the standard normal distribution table, the approximate probability corresponding to \( z = -0.36 \) is 0.3594 and for \( z = 0.71 \) is 0.7611. The probability of a value between 75.0 and 90.0 is \( 0.7611 - 0.3594 = 0.4017 \).

Step 4: Convert to Z-score for 75.0 and Less

Using the previously calculated z-score for 75.0, which is \( -0.36 \), check the standard normal table for the probability: 0.3594. This represents the probability of a value being 75.0 or less.

Step 5: Convert to Z-scores for 55.0 and 70.0

For 55.0, \( z = \frac{(55.0 - 80.0)}{14.0} = -\frac{25}{14} \approx -1.79 \). For 70.0, \( z = \frac{(70.0 - 80.0)}{14.0} = -\frac{10}{14} \approx -0.71 \).

Step 6: Find the Probability for 55.0 to 70.0

Using the standard normal distribution table, the probability for \( z = -1.79 \) is 0.0367 and for \( z = -0.71 \) is 0.2389. The probability of a value between 55.0 and 70.0 is \( 0.2389 - 0.0367 = 0.2022 \).

Key Concepts

Understanding the Z-score

The z-score is an important concept for understanding normal distributions. It tells us how far away a value is from the mean, measured in terms of the standard deviation. The formula for calculating the z-score is:
\[ z = \frac{(X - \mu)}{\sigma} \]
Here:

• \( X \) is the value from the dataset
• \( \mu \) is the mean of the dataset
• \( \sigma \) is the standard deviation

A z-score of 0 means the value is exactly at the mean, while a positive z-score indicates the value is above the mean. Conversely, a negative z-score means the value is below the mean. Z-scores allow us to compare values from different normal distributions, enabling us to determine probabilities drawn from these distributions by referring to a standard normal distribution table.

The standard normal distribution table, sometimes referred to as a z-table, provides the probability that a z-score is less than or equal to a particular value. This is crucial in probability calculations involving normal distributions.

Exploring Standard Deviation in Depth

Standard deviation is a measure that indicates the amount of variability or dispersion within a set of data values. In a normal distribution, a smaller standard deviation indicates that the data points are close to the mean, whereas a larger standard deviation implies a wider spread around the mean.

In the context of our exercise, the standard deviation is 14.0. This tells us how much individual data points spread out from the mean of 80.0. The role of standard deviation in the z-score formula is crucial, as it scales the difference between a data point and the mean. If you imagine a graph of a normal distribution curve:

• Approximately 68% of data falls within one standard deviation from the mean.
• About 95% is within two standard deviations.
• Nearly 99.7% lies within three standard deviations.

Understanding standard deviation gives us insight into how much "average" deviation from the mean exists in a dataset. This is essential when calculating probabilities and making predictions based on those probabilities.

Probability Calculation Simplified

Probability calculation in a normal distribution involves determining the likelihood of a data point falling within a specified range. By converting values to z-scores and referring to the standard normal distribution table, you can find these probabilities.

Let's break it down: 1. **Convert each data point of interest to a z-score.** This standardizes the values, allowing us to refer to a common framework (the z-table). 2. **Look up each z-score in the z-table.** This provides the cumulative probability up to that z-score. 3. **Calculate the difference between the cumulative probabilities of two z-scores to find the probability of them lying between these points.**
Using this method in practice:

• If you want the probability that a value is less than or equal to a certain amount, find its z-score and refer directly to the table.
• For a probability range, use two z-scores and subtract the cumulative probability of the lower from the higher score.

This framework simplifies complex probability calculations, making it easier to understand and apply.

Defining the Mean and Its Importance

The mean is a fundamental statistical measure representing the average of a dataset. It is calculated by summing all the values and dividing by the number of observations:
\[ \mu = \frac{\sum X}{n} \]
Where \(X\) are the data points and \(n\) is the number of observations. In probability and statistics, especially in the realm of normal distributions, the mean serves as the 'center' or the 'peak' of the distribution.

In our exercise, the mean is 80.0, which acts as the pivot point for understanding the spread (standard deviation) and calculated probabilities in the dataset. A key characteristic of the normal distribution is its symmetric bell-shaped curve centered around the mean.

• This symmetry implies equal spread to the right and left of the mean.
• Given a normal distribution, the mean equals both the median and mode.

Understanding the mean is vital, as it allows us to determine how individual data points (values) relate to the typical behavior of the dataset. This relationship is essential for making calculations like z-scores and probabilities in a normal distribution.