Problem 10 - Basic Statistics for Business and Economics

Question

The mean of a normal probability distribution is \(60 ;\) the standard deviation is \(5 .\) a. About what percent of the observations lie between 55 and \(65 ?\) b. About what percent of the observations lie between 50 and \(70 ?\) c. About what percent of the observations lie between 45 and \(75 ?\)

Step-by-Step Solution

Answer

a) 68% b) 95% c) 99.7%

Step 1: Understanding the Normal Distribution

We have a normal distribution with a mean \( \mu = 60 \) and a standard deviation \( \sigma = 5 \). We are to find the percentage of observations between certain values using the properties of a normal distribution.

Step 2: Use Empirical Rule for 55 to 65

The empirical rule states that about 68% of the data falls within one standard deviation of the mean in a normal distribution. Here, one standard deviation from the mean \( 60 \) is \([60 - 5, 60 + 5] = [55, 65]\). Thus, about 68% of the observations lie between 55 and 65.

Step 3: Use Empirical Rule for 50 to 70

Two standard deviations from the mean are \( \mu \pm 2\sigma = 60 \pm (2 \times 5) = [50, 70] \). The empirical rule suggests that about 95% of observations lie within two standard deviations. Thus, about 95% of the observations lie between 50 and 70.

Step 4: Use Empirical Rule for 45 to 75

Three standard deviations from the mean are \( \mu \pm 3\sigma = 60 \pm (3 \times 5) = [45, 75] \). The empirical rule indicates that about 99.7% of observations fall within three standard deviations. Therefore, about 99.7% of the observations lie between 45 and 75.

Key Concepts

Understanding the Empirical Rule

The empirical rule is a simple guideline to understand how data in a normal distribution are spread out around the mean. It's like a handy tool for quickly estimating the spread of values. In a normal distribution:

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

Knowing this rule helps predict the percentage of observations within specific intervals in a bell-shaped, normally distributed dataset.
For instance, if you're examining test scores or measuring product weights that follow a normal distribution, the empirical rule allows you to quickly find out how concentrated the observations are around the mean. This makes it a valuable tool for data analysis and statistics.

The Mean and Standard Deviation

The mean and standard deviation are central to understanding any normal distribution. The mean, often symbolized as μ, represents the average value of all the data points. It's the "center" of the normal distribution.
Imagine calculating the average score from a set of test results; that is what the mean tells you. On the other hand, the standard deviation, denoted by σ, measures the spread or dispersion of data around the mean.
Here's a simple way to think about it:

• If the standard deviation is small, the data are closely clustered around the mean, which implies less variability.
• A large standard deviation indicates that the data points are spread over a wider range, suggesting more variability.

In practical terms, having a good grasp of mean and standard deviation helps in predicting outcomes and understanding the distribution's shape and spread.

Percentage of Observations in Intervals

Using the empirical rule and knowing the mean and standard deviation, you can determine what percentage of observations fall within specific intervals. For example, if the mean is 60 and the standard deviation is 5:

• Between 55 and 65: This range represents one standard deviation from the mean. Based on the empirical rule, roughly 68% of observations fall within this range.
• Between 50 and 70: This interval covers two standard deviations. Here, about 95% of observations are included, as outlined by the empirical rule.
• Between 45 and 75: Spanning three standard deviations, this interval contains about 99.7% of observations.

Understanding these percentages helps in making informed decisions in fields like quality control, where predicting product consistency within certain bounds is crucial. It's about seeing where most of the data falls within a normal distribution.