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Chapter 3: Problem 42

Weights aren't normal. The heights of people of the same sex and similar ages follow a Normal distribution reasonably closely. Weights, on the other hand, are not Normally distributed. The weights of women aged 20-29 in the United States have mean \(161.9\) pounds and median \(149.4\) pounds. The first and third quartiles are \(126.3\) pounds and \(181.2\) pounds, respectively. What can you say about the shape of the weight distribution? Why?

Short Answer

Expert verified
The weight distribution is positively skewed because the mean is greater than the median.

Step by step solution

01

Understanding Skewness

The mean of the weight distribution is given as 161.9 pounds, and the median is 149.4 pounds. When the mean is greater than the median, it suggests that the data is positively skewed.
02

Interpreting Quartiles

The given first quartile (Q1) is 126.3 pounds, and the third quartile (Q3) is 181.2 pounds. The interquartile range (IQR) is calculated as Q3 - Q1, which is 54.9 pounds.
03

Assessing Distribution Shape

Considering the difference between the mean and median, along with the spread measured by the interquartile range, it suggests that the distribution of weights has a positive skewness, where there are some higher values pulling the mean above the median.

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Key Concepts

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

Normal Distribution
A normal distribution is a bell-shaped statistical concept that shows how data values are spread out evenly on both sides of a central mean. This concept is important in many fields because it describes how values tend to cluster around the average.
In a normally distributed set of data:
  • The mean, median, and mode are equal and located at the center.
  • Data symmetrically spreads around the mean.
  • About 68% of data falls within one standard deviation from the mean, 95% within two, and 99.7% within three.
Weights don鈥檛 always follow a normal distribution, as seen in this problem. Instead, other factors, like skewness, affect how the data is distributed.
Skewness
Skewness refers to the asymmetry or lack of symmetry in a distribution of data points. A dataset is considered positively skewed if it stretches more on the right side, where the tail is longer on this side than on the left.
Key signs of positive skewness in a distribution include:
  • Mean greater than the median鈥攊ndicative of long right tail.
  • Concentration of data points on the left.
In the weights example, the mean weight is 161.9 pounds while the median is 149.4 pounds. This gap indicates positive skewness, implying that higher weights are pulling the mean to be more than the median.
Interquartile Range
The interquartile range (IQR) is a measure of statistical dispersion, representing the range where the middle 50% of the data lies. You find it by subtracting the first quartile (Q1鈥攚here 25% of data is below this value) from the third quartile (Q3鈥攚here 75% of data is below).
To illustrate:
  • For the weights data: IQR = Q3 - Q1 = 181.2 - 126.3 = 54.9 pounds.
This range provides insight into how spread out the central group of data is. A larger IQR indicates a more spread-out distribution. The example shows a moderate spread of weights around the central values, supporting the skewness observation.
Mean and Median
Mean and median are both measures of central tendency, but they cannot always be used interchangeably. Understanding the difference is crucial.
  • Mean: The average of all data points, useful for normally distributed data.
  • Median: The middle value when data is ordered from smallest to largest, useful in skewed distributions to give a clearer central value.
In the weights problem, the mean is higher than the median which indicates skewness. So, recognizing whether a dataset skews can help determine which measure provides a clearer picture of the data鈥檚 central tendency.

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Most popular questions from this chapter

The proportion of observations from a standard Normal distribution that take values between 1 and 2 is about (a) \(0.025\). (b) \(0.135 .\) (c) \(0.160\).

The scores of adults on an IQ test are approximately Normal with mean 100 and standard deviation 15. Alysha scores 135 on such a test. She scores higher than what percent of all adults? (a) About \(5 \%\) (b) About \(95 \%\) (c) About \(99 \%\)

What's your percentile? Reports on a student's test score such as the SAT or a child's height or weight usually give the percentile as well as the actual value of the variable. The percentile is just the cumulative proportion stated as a percent: the percent of all values of the variable that were lower than this one. The upper arm lengths of females in the United States are approximately Normal with mean \(35.8 \mathrm{~cm}\) and standard deviation \(2.1 \mathrm{~cm}\), and those for males are approximately Normal with mean \(39.1 \mathrm{~cm}\) and standard deviation \(2.3 \mathrm{~cm}\). (a) Cecile, a 73-year-old female in the United States, has an upper arm length of \(33.9 \mathrm{~cm}\). What is her percentile? (b) Measure your upper arm length to the nearest tenth of a centimeter, referring to Exercise \(3.5\) (page 84 ) for the measurement instructions. What is your arm length in centimeters? What is your percentile?

Sketch Density Curves. Sketch density curves that describe distributions with the following shapes: (a) Symmetric, but with two peaks (that is, two strong clusters of observations) (b) Single peak and skewed to the left

Fruit flies. The common fruit fly Drosophila melanogaster is the most studied organism in genetic research because it is small, is easy to grow, and reproduces rapidly. The length of the thorax (where the wings and legs attach) in a population of male fruit flies is approximately Normal with mean \(0.800\) millimeter (mm) and standard deviation \(0.078 \mathrm{~mm}\). (a) What proportion of flies have thorax length less than \(0.7 \mathrm{~mm}\) ? (b) What proportion have thorax length greater than \(1.0 \mathrm{~mm}\) ? (c) What proportion have thorax length between \(0.7 \mathrm{~mm}\) and \(1.0 \mathrm{~mm}\) ?
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