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Chapter 3: Problem 15

Which of these variables is most likely to have a Normal distribution? a. Income per person for 150 different countries b. Sale prices of 200 homes in Santa Barbara, California c. Lengths of 100 newborns in Connecticut

Short Answer

Expert verified
Lengths of newborns are most likely to have a normal distribution.

Step by step solution

01

Understanding Uniformity and Symmetry

The normal distribution often models data that arises naturally or in a context where most values are around a central median, with symmetrical spread. Consider biological measurements or other variables that naturally balance around an average.
02

Analysis of Income per Person

Income per person for 150 different countries is likely to be skewed because of disparities in wealth between different countries. A few countries typically have disproportionately high incomes, and many have much lower incomes, leading to a right-skewed distribution.
03

Analysis of Sale Prices of Homes

Sale prices of homes can also be highly skewed. The prices are influenced by various factors, such as location and size, and may have a significant number of extreme high or low values, making the distribution likely non-normal.
04

Analysis of Lengths of Newborns

Lengths of newborns are biological measurements. These often follow a normal distribution because they tend to cluster around a median value, with few exceptionally long or short measurements. These characteristics suggest it is the most likely to be normally distributed.

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Key Concepts

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

Skewness
Skewness refers to the measure of asymmetry in the distribution of data. When data is perfectly symmetric, it is likely to follow a normal distribution. However, skewness indicates a departure from this symmetry.
In a skewed distribution, data is biased towards the left or the right:
  • Right-skewed (or positively skewed) has a long tail on the right side. Examples include income or sales data, where a small number of observations are significantly higher than the rest.
  • Left-skewed (or negatively skewed) has a long tail on the left side. This pattern is less common but can occur in exams where most students score high marks, leaving a few low outliers.
In essence, the skewness of a data set tells us how data points fare compared to the average. It helps understand the tendency for data points to skew around either the high or low end, as seen in income distributions or home sale prices.
Biological Measurements
Biological measurements often follow a normal distribution. This is because they tend to cluster around a central, typical value naturally. Imagine measuring the heights of a population of adults; most individuals will be around the average height, with fewer being exceptionally short or tall.
Their characteristics include:
  • Symmetry around a central peak (bell-shaped curve).
  • A concentration of data points towards the mean, with probabilities tapering off as you move away from the center.
  • Consistency, where measurements are subject to biological norms rather than external extremes.
Lengths of newborns, a typical biological measurement, follow this pattern as they usually exhibit little variation and any exceptions to the average are minimal.
Central Tendency
Central tendency refers to the measure of the "center" of a dataset. It is used to locate where most of the data points are concentrated. The most common measures of central tendency are:
  • Mean, which is the arithmetic average of all data points.
  • Median, which is the middle value when the data is ordered.
  • Mode, which is the most frequently occurring value.
In a normal distribution, these three measures coincide at a single point that represents the peak of the curve.
Central tendency is invaluable in understanding how data is clustered. It helps in predicting patterns and making comparisons across different datasets. For example, lengths of newborns tend to show remarkably consistent central tendencies across different groups which suggests data is normally distributed, unlike income or home sale prices.

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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. \(0.160\).

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 \(1.0 \mathrm{~mm}\) ?

The distribution of hours of sleep per weeknight among college students is found to be Normally distributed, with a mean of \(6.5\) hours and a standard deviation of 1 hour. What range contains the middle \(95 \%\) of hours slept per weeknight by college students? a. \(5.5\) and \(7.5\) hours per weeknight b. \(4.5\) and \(7.5\) hours per weeknight c. \(4.5\) and \(8.5\) hours per weeknight

Monsoon Rains. The summer monsoon rains bring \(80 \%\) of India's rainfall and are essential for the country's agriculture. Records going back more than a century show that the amount of monsoon rainfall varies from year to year according to a distribution that is approximately Normal, with mean 852 millimeters \((\mathrm{mm})\) and standard deviation 82 \(\mathrm{mm} .{ }^{3}\) Use the 68-95-99.7 rule to answer the following questions. a. Between what values do the monsoon rains fall in the middle \(95 \%\) of all years? b. How small are the monsoon rains in the driest \(2.5 \%\) of all years?

Acid Rain? Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by \(\mathrm{pH}\) on a scale of 0 to 14 . Distilled water has \(\mathrm{pH}\) 7.0, and lower \(\mathrm{pH}\) values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below \(5.0\). The \(\mathrm{pH}\) of rain at one location varies among rainy days according to a Normal distribution with mean \(5.43\) and standard deviation \(0.54\). What proportion of rainy days have rainfall with \(\mathrm{pH}\) below \(5.0\) ?
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