Question

A distribution of measurements is relatively mound-shaped with a mean of 50 and a standard deviation of \(10 .\) Use this information to find the proportion of measurements in the intervals given in Exercises \(6-11 .\) 40 or more

Short answer

Answer: The proportion of measurements in the interval of 40 or more is approximately 97.5%.

Step-by-step solution

Identify the mean and standard deviation

The mean (µ) is 50 and the standard deviation (σ) is 10.

Apply the Empirical Rule

According to the Empirical Rule: - 68% of measurements lie between (µ - σ) and (µ + σ), which is between 40 and 60. - 95% of measurements lie between (µ - 2σ) and (µ + 2σ), which is between 30 and 70. - 99.7% of measurements lie between (µ - 3σ) and (µ + 3σ), which is between 20 and 80.

Find the proportion of measurements 40 or more

Since 68% of measurements lie between 40 and 60, we need to find the proportion of measurements that are either between 40 and 50 or more than 50. Half of the measurements lying between 40 and 60 will be between 40 and 50. Thus, 34% of measurements must be in this range (since 68% / 2 = 34%). As 95% of measurements lie between 30 and 70, this means that 5% lie outside this range. Since the distribution is symmetrical, 2.5% of measurements will be above 70 (since 5% / 2 = 2.5%). To find the proportion of measurements that are 40 or more, we simply add these percentages together: 34% (between 40 and 50) + 100% - 2.5% (above 70) = 97.5%. The proportion of measurements in the interval of 40 or more is approximately 97.5%.

Key concepts

Normal Distribution

A normal distribution, often referred to as the bell curve, is a probability distribution that is symmetric around the mean. It represents how the values of a dataset are distributed. Imagine a dataset where most of the values cluster around a central point, with fewer values tapering off symmetrically on both sides. This central point is the mean, and the spread of the data is characterized by the standard deviation.

• The shape is symmetric and bell-shaped.
• Most data points lie close to the mean, emphasizing the 'mound' shape.
• It is defined by two key parameters: mean (the central tendency) and standard deviation (the spread or dispersion).

In practical terms, understanding normal distribution helps us to make predictions about data behavior and probability, as seen in various fields such as biology, finance, and social sciences.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a set of data values. In the context of a normal distribution, it tells us how far away the data points typically are from the mean.

• A small standard deviation indicates that data points are clustered tightly around the mean.
• A large standard deviation suggests a wider spread of data points.
• In a normal distribution, the standard deviation helps us understand intervals within which a certain percentage of the data lies.

For example, using the Empirical Rule, we know that approximately 68% of the data will fall within one standard deviation from the mean — this is a powerful tool for making predictions and understanding data variability.

Mean

The mean, often called the average, is the sum of all data points divided by the number of data points. It provides a central value around which the data is distributed. Think of the mean as a balancing point for the entire dataset.

• It represents the central tendency of the data.
• In a normal distribution, the mean is at the center of the symmetric bell curve.
• Affects the position of the normal distribution on the horizontal axis (x-axis).

When we have a normal distribution, the mean defines where the peak of the curve is. It is crucial in determining various statistical properties and understanding how data is expected to behave under normal conditions.

Proportion Analysis

Proportion analysis in statistics involves determining what portion of the data falls within certain limits. This is particularly useful when analyzing data with a normal distribution to understand probabilities.

• Uses percentages to represent data falling within intervals.
• The Empirical Rule helps estimate the proportion of data within one, two, or three standard deviations from the mean.
• In the exercise, we saw how to calculate the proportion of measurements that exceeded a specified value.

In the given exercise, when searching for the percentage of measurements that are 40 or more, we applied the Empirical Rule and basic arithmetic to determine that approximately 97.5% of the data points meet this criterion. This type of analysis is an essential skill for interpreting statistical data effectively.