Question

Heights of adult women in the United States are normally distributed with a population mean of \(\mu=63.5\) inches and a population standard deviation of \(\sigma=\) 2.5. A medical re- searcher is planning to select a large random sample of adult women to participate in a future study. What is the standard value, or z-value, for an adult woman who has a height of 68.5 inches?

Short answer

The z-value for a height of 68.5 inches is 2.

Step-by-step solution

Understand the Problem

The problem requires us to calculate the z-value for an adult woman who has a height of 68.5 inches. We know that the heights are normally distributed with a mean (\(\mu\)) of 63.5 inches and a standard deviation (\(\sigma\)) of 2.5 inches.

Identify the Z-value Formula

The z-value formula, which helps us determine how many standard deviations an element is from the mean, is given by:\[ z = \frac{X - \mu}{\sigma} \]where \(X\) is the value of the element (height in this case), \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.

Substitute the Values

In the z-value formula, substitute \(X = 68.5\) inches, \(\mu = 63.5\) inches, and \(\sigma = 2.5\) inches:\[ z = \frac{68.5 - 63.5}{2.5} \]

Calculate the Z-value

Perform the calculation:\[ z = \frac{68.5 - 63.5}{2.5} = \frac{5}{2.5} = 2 \]The z-value is 2.

Key concepts

Normal Distribution

A normal distribution is a statistical term used to describe how data points are spread out in a dataset. Imagine a symmetrical bell-shaped curve – that's what a normal distribution looks like. This type of distribution is very important in statistics because it allows us to make inferences about the data. It's characterized by two main things: the mean and the standard deviation.

In a normal distribution, most data points are close to the mean, and fewer data points are further away in either direction. This means that in the middle of the curve, you find the highest frequency of occurrences.

• The peak of the bell curve represents the mean, where most occurrences are concentrated.
• The tails of the curve taper off symmetrically, indicating lesser occurrence as you move further from the mean.

This meaningful pattern allows researchers to predict and understand trends and behaviors within a dataset. Because many natural phenomena like height, IQ scores, and measurement errors tend to be normally distributed, the normal distribution model is widely used across various fields.

Standard Deviation

Standard deviation is a measure that tells us how spread out the numbers in a data set are. When the data points cluster closely around the mean, the standard deviation is low. Conversely, if they spread out over a wider range, the standard deviation is higher.

In our exercise, the standard deviation of adult women's heights is 2.5 inches. This figure helps us understand the degree of variation among those heights.

• A low standard deviation indicates that most women's heights are close to the average height of 63.5 inches.
• A higher standard deviation would mean that the heights are more varied.

Understanding standard deviation is key as it provides insight into the consistency of a particular trait or measure within a population. In practical terms, it helps in assessing reliability and forecasting expectations throughout various statistical assessments.

Mean

The mean, often referred to as the average, is a central value that summarizes a set of numbers. Calculating the mean involves adding up all the numbers and then dividing by the quantity of numbers. In our context, the mean height of adult women is 63.5 inches.

The mean provides a quick snapshot of the typical value in the data set. However, it doesn't convey information about the spread or variance of the data. This is why it's often paired with the standard deviation.

• In the normal distribution, the mean is the central point where the distribution curve peaks.
• Understanding the mean is vital for making sense of data patterns and forming expectations around typical values.

The mean serves as a foundational element of many statistical analyses and is among the simplest yet most informative measures in statistics.

Standard Score

A standard score, commonly known as the z-score, tells us where a particular data point stands in comparison to the mean of the dataset in terms of standard deviations. In simple terms, it helps determine how typical or atypical a value is in the context of the whole. In our exercise, the z-score calculation for a woman who is 68.5 inches tall, given the population mean of 63.5 inches and a standard deviation of 2.5 inches, is 2.

Here's how it works:

• The formula for calculating a z-score is: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the individual data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
• A z-score of 2 indicates that the woman's height is two standard deviations above the mean height.

Z-scores are incredibly useful as they allow comparisons across different data sets or different units of measurement by standardizing the data. They help in identifying how extreme or unusual a data point is and are widely used in research and decision-making processes.