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 \(6-11 .\) Less than 60

Short answer

Answer: Approximately 84.13% of the measurements are less than 60.

Step-by-step solution

Calculate the Z-Score

To find the proportion of measurements less than 60, we first need to calculate the Z-score. The Z-score is the number of standard deviations a data point is away from the mean. The formula for calculating the Z-score is: \(Z = \frac{X - \mu}{\sigma}\) where X is the value we want to find the proportion for (60), \(\mu\) is the mean (50), and \(\sigma\) is the standard deviation (10). \(Z = \frac{60 - 50}{10} = 1\)

Use the Standard Normal Distribution Table

Now that we have the Z-score, we can use the standard normal distribution table (also known as the Z-table) to find the proportion of measurements less than 60. The table gives the area under the curve to the left of the Z-score. Look up the Z-score of 1 in the Z-table. If we check a standard normal distribution table, we find that a Z-score of 1 corresponds to a probability of 0.8413.

Interpret the Result

The probability we found, 0.8413, represents the proportion of measurements that fall below a value of 60 in our mound-shaped distribution. This means that approximately 84.13% of the measurements are less than 60.

Key concepts

Standard Deviation

In statistics, standard deviation is a measure that tells us how spread out numbers are around the mean (average) of a data set. Think of it as a way to capture how much variation there is from the average. A low standard deviation means that most numbers are close to the mean, while a high standard deviation means that numbers are more spread out.

For example, if we have a group of test scores with a mean of 70 and a standard deviation of 5, most scores are expected to be relatively close to 70, within a range of 65 to 75. In the context of our exercise, with a mean of 50 and a standard deviation of 10, the interval from 40 to 60 would cover most of the data points, signifying a moderately wide spread around the mean.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a bell-shaped curve that is symmetrical about the mean. Most values cluster around a central region and probabilities for values taper off as you move further from the mean in both directions. Real-world examples often approximate a normal distribution, such as heights of people, test scores, or measurement errors.

When a dataset follows a normal distribution, we use certain properties to calculate probabilities and make predictions. For instance, about 68% of values are within one standard deviation from the mean, 95% are within two standard deviations, and nearly all values are within three standard deviations. This property is crucial in determining probabilities for specific intervals around the mean.

Probability

Probability is the measure of how likely an event is to occur. It ranges from 0 (impossible) to 1 (certainty). In the context of standard deviations and normal distributions, we often ask questions like 'What is the probability that a measurement falls within a particular range?'

The concept of Z-scores and the standard normal distribution table, sometimes called a Z-table, help in answering these questions. By converting raw scores into Z-scores, we standardize the data and refer to the table where probabilities are listed for us. In the exercise example, we converted the score of 60 into a Z-score and then consulted the Z-table to determine that the probability of measurements being less than 60 was about 84.13%.

Statistical Measurements

Statistical measurements, often called descriptive statistics, include different ways to summarize and describe the central tendency, variation, and distribution shape of a dataset. Central tendency measures include the mean, median, and mode. Variation is often described using range, variance, or standard deviation.

The Z-score is another example of a statistical measurement that not only indicates how many standard deviations an element is from the mean but also allows for comparison across different datasets by normalizing the data. Understanding these statistical concepts is critical when interpreting data, as they give us a deeper insight into the characteristics of our data and enable us to make informed decisions based on statistical evidence.