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Chapter 3: Problem 31

The dotplot shows heights of college women; the mean is 64 inches \((5\) feet 4 inches), and the standard deviation is 3 inches. a. What is the \(z\) -score for a height of 58 inches ( 4 feet 10 inches)? b. What is the height of a woman with a \(z\) -score of \(1 ?\)

Short Answer

Expert verified
The z-score for a height of 58 inches is -2, and the height of a woman with a z-score of 1 is 67 inches.

Step by step solution

01

Calculate the z-score

The z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating z-score is \( z = \frac{(X - \mu)}{\sigma} \) where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In this problem, \( X = 58 \) inches, \( \mu = 64 \) inches, and \( \sigma = 3 \) inches. Substituting these values into the formula will give us the z-score for a height of 58 inches.
02

Compute z-score for 58 inches

Substitute values into the z-score formula: \( z = \frac{(58 - 64)}{3} \) which simplifies to \( z = -2 \).
03

Calculate the height for a z-score of 1

To find the height corresponding to a given z-score, we rearrange the z-score formula to solve for \( X \): \( X = z*\sigma + \mu \). In this problem, \( z = 1 \), \( \sigma = 3 \) inches, and \( \mu = 64 \) inches. Substituting these values into the formula gives the height of a woman with a z-score of 1.
04

Compute height for z-score 1

Substitute values into the formula: \( X = 1*3 + 64 \), which simplifies to \( X = 67 \) inches. This means a woman with a z-score of 1 has a height of 67 inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Standard Deviation
Standard deviation is a critical statistical tool that helps us determine the amount of variation or dispersion in a set of data. In simpler terms, it indicates how much the data points differ from the average value, which we call the mean.

When you hear 'standard deviation,' think about average distance. Imagine everyone standing in a group at varying distances from a marker. The distances from that marker to each person vary slightly. Standard deviation is somewhat like calculating an average of those varying distances.

  • A small standard deviation indicates that the data points are close to the mean.
  • A larger standard deviation suggests that the data points are more spread out over a wider range of values.
In our exercise, the standard deviation is 3 inches, which tells us how much the heights of college women vary from the mean height of 64 inches.
Exploring the Mean
The mean is simply the average of all numbers in a data set and is calculated by summing all the values and then dividing by the count of values. It's an essential measure in statistics because it provides a central value to which data points can be compared.

To calculate the mean from a list:
  • First, add up all the individual numbers in your data set.
  • Then, divide this sum by the total number of data points you have.
In the context of our exercise, we are given the mean height of college women, which is 64 inches. This means if you sum all the heights and then divide by the number of women, their average height is 64 inches. Understanding the mean is crucial, as it serves as a reference point in calculating z-scores鈥攈ow much a particular value deviates from this central average measure.
Visualizing Data with a Dotplot
A dotplot is an essential tool for visually displaying the distribution of data points. Each dot represents a single observation or measurement within the data set, and they are arranged along a numerical axis.

Dotplots are beneficial because:
  • They provide a clear visualization of data distribution and frequency.
  • They help in quickly identifying clusters, gaps, and outliers in the data set.
In our exercise, a dotplot is used to represent the heights of college women. By plotting each height with a dot on a line, we can quickly see how these heights are spread across different values. This kind of visual representation helps in understanding the general distribution and helps connect visual perception with numerical measures like the mean and standard deviation.

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Most popular questions from this chapter

An exam has a mean of 70 and a standard deviation of \(10 .\) What exam score corresponds to a z-score of \(1.5 ?\)

Babies born after 40 weeks gestation have a mean length of \(52.2\) centimeters (about \(20.6\) inches). Babies born one month early have a mean length of \(47.4 \mathrm{~cm}\). Assume both standard deviations are \(2.5 \mathrm{~cm}\) and the distributions are unimodal and symmetric. (Source: www.babycenter.com) a. Find the standardized score (z-score), relative to all U.S. births, for a baby with a birth length of \(45 \mathrm{~cm}\). b. Find the standardized score of a birth length of \(45 \mathrm{~cm}\) for babies born one month early, using \(47.4\) as the mean. c. For which group is a birth length of \(45 \mathrm{~cm}\) more common? Explain what that means.

When you are comparing two sets of data, and one set is strongly skewed and the other is symmetric, which measures of the center and variation should you choose for the comparison?

3.64 Passing the Bar Exam The dotplot shows the distribution of passing rates for the bar exam at 185 law schools in the United States in 2009 . The five number summary is $$ 26,80,86,90,100 $$ Draw the boxplot and explain how you determined where the whiskers go.

a. In your own words, describe to someone who knows only a little statistics how to recognize when an observation is an outlier. What action(s) should be taken with an outlier? b. Which measure of the center (mean or median) is more resistant to outliers, and what does "resistant to outliers" mean?
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