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 \(6-11 .\) Between 30 and 60

Short answer

Answer: The approximate proportion of measurements in the interval between 30 and 60 is 84%.

Step-by-step solution

Identify the interval given

The given interval is between 30 and 60, which means we need to find the proportion of measurements that fall within this range.

Identify the mean and standard deviation

The mean of the distribution is 50, and the standard deviation is 10.

Apply the empirical rule

Since the distribution is mound-shaped, we can use the empirical rule to approximate the proportion of measurements in the given interval. Recall that the empirical rule states that about 68% of the data falls within one standard deviation of the mean in a normal distribution. One standard deviation below the mean is 50 - 10 = 40, and one standard deviation above the mean is 50 + 10 = 60. Thus, the interval between 30 and 60 encompasses the interval of one standard deviation below and above the mean (between 40 and 60) plus the additional 10 points difference between 30 and 40.

Calculate the proportion

Since we know that about 68% of the measurements fall within one standard deviation of the mean (between 40 and 60), we need to account for the remaining interval between 30 and 40. Assuming a symmetrical distribution, we can estimate that half of the remaining 32% of the data would fall below 40 and half above 60. Thus, about 16% of the data would fall between 30 and 40.

Find the total proportion in the given interval

The proportion of measurements between 30 and 60 can be found by adding the proportions between 30 and 40 (16%) and 40 and 60 (68%). So, the proportion of measurements between 30 and 60 is 16% + 68% = 84%. The proportion of measurements in the interval between 30 and 60 is approximately 84%.

Key concepts

Normal Distribution

In statistics, a normal distribution is a fundamental concept, characterized by a symmetrical, bell-shaped curve. This curve represents the distribution of a set of data points. The most important feature of a normal distribution is that it is perfectly symmetrical about the mean, meaning the data is equally distributed on both sides.

Normal distributions are commonly seen in natural and social sciences. It describes how items are spread around a central value, with fewer items at the extremes. This distribution is defined by two parameters: the mean and the standard deviation, which determine the center and width of the curve, respectively.

• It provides a useful model because many real-world variables such as heights, test scores, and measurement errors tend to follow this pattern.
• The empirical rule, which we'll discuss later, relies on the properties of the normal distribution to make estimations about data spread.

Mean and Standard Deviation

The mean and standard deviation are two fundamental statistics that describe a distribution.

The **mean** is the average of all data points. It's calculated by adding up all the numbers and then dividing by the total count. In a normal distribution, the mean represents the center of the distribution.

The **standard deviation** measures the spread of the data. It's the average distance of each data point from the mean. A small standard deviation indicates data points are close to the mean, while a large standard deviation implies greater spread. Together, the mean and standard deviation provide a complete picture of the data's distribution.

• They help in understanding the overall pattern and in predicting outcomes for normally distributed data.
• In the empirical rule, they are crucial for calculating intervals.

Proportion Calculation

Proportion calculation in the context of normal distribution often involves the empirical rule. This rule, also known as the 68-95-99.7 rule, provides estimates of data distribution within standard deviations from the mean.

Here's how it works:

• About 68% of data falls within one standard deviation ( ) of the mean.
• Approximately 95% is within two standard deviations ( ).
• Almost 99.7% falls within three standard deviations ( ).

Applying this rule allows you to calculate the proportion of measurements within a particular interval. For example, if you need to find the proportion of data between two points on a normal curve, you apply the empirical rule to estimate the data present in those intervals.

Using the example of measurements between 30 and 60, the interval within one standard deviation from the mean provided 68%. By considering the symmetrical nature of normal distribution, additional calculations give you the total proportion, which in this case was 84%.