Question

A distribution of measurements has a mean of 75 and a standard deviation of \(5 .\) You know nothing else about the size or shape of the data. Use this information to find the proportion of measurements in the intervals given. Between 65 and 85

Short answer

Answer: The proportion of measurements in the interval between 65 and 85 is 0.9544, or 95.44%.

Step-by-step solution

Identify the given values

In this problem, the mean μ is 75, the standard deviation σ is 5, and the interval in which we want to find the proportion is [65, 85].

Calculate the z-scores for the given interval

The formula for calculating the z-score is $$z = \frac{x - μ}{σ}.$$ For the lower bound of the interval (65), we will calculate the z-score like this: $$z_1 = \frac{65 - 75}{5} = -2.$$ For the upper bound of the interval (85), we will calculate the z-score like this: $$z_2 = \frac{85 - 75}{5} = 2.$$ The z-scores for the interval [65, 85] are -2 and 2.

Use the standard normal table to find the proportion

Using a standard normal table, we will look up the values for our z-scores and find the area under the curve to the left of each z-score. For z = -2, the area to the left is 0.0228. For z = 2, the area to the left is 0.9772. Now, we will subtract the area corresponding to z = -2 from the area corresponding to z = 2: $$0.9772 - 0.0228 = 0.9544.$$

Interpret the result

The proportion of measurements in the interval between 65 and 85 in the given data distribution is 0.9544, or 95.44%.

Key concepts

Z-Score Calculation

Understanding the z-score is crucial when working with normal distributions. In simple terms, a z-score represents how many standard deviations a particular data point is from the mean. The formula for z-score calculation is:
\[ z = \frac{x - \mu}{\sigma} \]
where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Finding the z-scores allows us to determine the relative position of a data point within a distribution. For example, if a z-score is positive, the data point lies above the mean, and if it's negative, it falls below the mean. Teaching this concept should focus on the application of the formula and the interpretation of z-scores. Students can practice calculating z-scores for various data points and learning what these scores indicate about the data's position in the distribution.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean of the data set, while a high standard deviation indicates that the values are spread out over a wider range. It is represented by the Greek letter sigma (\( \sigma \)).
Standard deviation plays a pivotal role in understanding data consistency. For any data set or population, computing the standard deviation provides us with a numerical value that helps us understand the variability. To ensure students grasp this concept, emphasis should be placed on calculating standard deviation, recognizing its significance in the context of the data, and how it affects the shape of the normal distribution.

Normal Distribution

A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The shape of the normal distribution is often referred to as a 'bell curve.'
In normal distribution, the mean, median, and mode are all the same, and the distribution is determined by two parameters, the mean (\( \mu \)) and the standard deviation (\( \sigma \)). The area under the curve in a normal distribution represents probability, and the total area under the curve adds up to one. Visualizing the bell curve can be instrumental in helping students understand the normal distribution. Real-world examples, such as grades or heights, can be practical for showing the applicability of the normal distribution.

Proportion of Measurements

The proportion of measurements within a given range in a normal distribution is a critical concept in statistics. It tells us what fraction of the data falls within a certain interval. The problem provided illustrates how to calculate the proportion of measurements between two points using the z-scores for the boundaries of that range.
By converting our interval to z-scores and using a standard normal table (or a calculator with the normal distribution function), we find the area under the curve between those z-scores, which represents the proportion. This area, or probability, can be used to predict the likelihood of a measurement falling within that range. It's important to teach students how to read standard normal tables and understand the significance of the proportion in the context of data analysis. Hands-on practice with diverse scenarios will reinforce the knowledge of how to determine the proportion of measurements in any given dataset.