Question

A set of data has a mean of 75 and a standard deviation of \(5 .\) You know nothing else about the size of the data set or the shape of the data distribution. a. What can you say about the proportion of measurements that fall between 60 and \(90 ?\) b. What can you say about the proportion of measurements that fall between 65 and \(85 ?\) c. What can you say about the proportion of measurements that are less than \(65 ?\)

Short answer

Question: Using the Empirical Rule and the given mean (75) and standard deviation (5), estimate the proportions of measurements within the following ranges: a. Between 60 and 90 b. Between 65 and 85 c. Less than 65

Step-by-step solution

Evaluate A: Proportion between 60 and 90

Using the Empirical Rule, we will be comparing 60 and 90 to the given mean (75) and standard deviation (5). First, compute the range in terms of standard deviations: 60 is 3 standard deviations below the mean (15 below the mean divided by 5), so \(60 = 75 - 3(5)\). 90 is 3 standard deviations above the mean (15 above the mean divided by 5), so \(90 = 75 + 3(5)\). Since both 60 and 90 are 3 standard deviations away from the mean, we can say that about 99.7% of measurements fall between this range.

Evaluate B: Proportion between 65 and 85

Now, we are looking at measurements between 65 and 85. Compute the range in terms of standard deviations: 65 is 2 standard deviations below the mean (10 below the mean divided by 5), so \(65 = 75 - 2(5)\). 85 is 2 standard deviations above the mean (10 above the mean divided by 5), so \(85 = 75 + 2(5)\). Since both 65 and 85 are 2 standard deviations away from the mean, we can say that about 95% of measurements fall between this range.

Evaluate C: Proportion less than 65

We will be analyzing the proportion of measurements less than 65. We already determined that 65 is 2 standard deviations below the mean in the previous part. Since about 95% of measurements fall within 2 standard deviations of the mean, the remaining 5% must be split evenly between the tails. Therefore, approximately 2.5% of measurements are less than 65. Answer: a. About 99.7% of measurements fall between 60 and 90. b. About 95% of measurements fall between 65 and 85. c. About 2.5% of measurements are less than 65.

Key concepts

Mean

The mean is the simple average of a set of numbers. It is one way of defining the "center" of your data, and it provides you with a basic understanding of where the data points are concentrated.

• To calculate the mean, you sum up all the data points and divide by the number of points.
• The mean is a measure of central tendency, indicating the typical value in a data set.

In our exercise, the mean is given as 75, which suggests that, on average, the values in this data set hover around this mark. The mean helps us apply the Empirical Rule, which requires knowing the center or average of the data.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of numbers. A low standard deviation means the data points are close to the mean, while a high standard deviation means they are spread out over a larger range of values.

• It provides insight into the consistency of the data.
• Helps in understanding the spread or "spread-out" nature of the data around the mean.

In our current context, the standard deviation is 5. This tells us how much, on average, the data points deviate from the mean of 75. By knowing this, we can use the Empirical Rule to predict proportions of data within certain ranges of standard deviations from the mean.

Normal Distribution

A normal distribution, often referred to as a "bell curve," is a probability distribution that is symmetric around its mean. Most of the data points cluster around the central peak, and probabilities for values taper off equally in both directions from the mean.

• It is defined by its mean and standard deviation.
• The empirical rule can be applied effectively if data is normally distributed.
• This rule states that about 68% of data falls within one standard deviation, about 95% within two, and nearly 99.7% within three.

In this exercise, even though we don't know if the data is normally distributed, we assume for the purpose of applying the empirical rule, which provides us approximations for the proportions of data.

Probability

Probability is a way to quantify uncertainty, representing how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain).

• Used to draw conclusions about what proportion of data falls within a particular range.
• Associated with the area under a normal distribution curve for specified bounds.

In this data exercise, probability helps us estimate the proportion of measurements that fall within certain ranges using the Empirical Rule. For instance, we calculate the probability of data falling between certain values, say 60 and 90, knowing the mean and standard deviation, assuming normality.