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Chapter 3: Problem 31

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 \(5.0\) ?

Short Answer

Expert verified
About 21.3% of rainy days have rainfall with pH below 5.0.

Step by step solution

01

Understand the Problem

We need to find the proportion of rainy days with a pH below 5.0, given that the pH varies normally with a mean of 5.43 and a standard deviation of 0.54.
02

Standardize the Value

Convert the pH value of 5.0 into a z-score using the formula \( z = \frac{X - \mu}{\sigma} \), where \( X \) is 5.0, \( \mu \) is 5.43, and \( \sigma \) is 0.54. This gives us:\[ z = \frac{5.0 - 5.43}{0.54} \approx -0.796 \]
03

Use the Standard Normal Distribution

Look up the z-score \(-0.796\) in the standard normal distribution table to find the probability that the z-score is less than this value. From the table or using a calculator, \( P(Z < -0.796) \approx 0.213 \).
04

Interpret the Result

The probability 0.213 can be interpreted as the proportion of rainy days with a pH below 5.0. Thus, approximately 21.3% of rainy days are defined as having acid rain.

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Key Concepts

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

pH Scale
The pH scale is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It ranges from 0 to 14. Here's a simple breakdown:
  • A pH value of 7 is considered neutral, which is the pH of pure distilled water.
  • Values less than 7 indicate acidity, with lower numbers suggesting higher acidity.
  • Values greater than 7 indicate alkalinity or basicity.
Each unit change on the pH scale represents a tenfold change in concentration of hydrogen ions \( H^+ \). This means that a solution with a pH of 4 is ten times more acidic than one with a pH of 5. Understanding the pH scale is key in environmental science to assess the potential impact of substances like acid rain on ecosystems.
Acid Rain
Acid rain refers to precipitation that is more acidic than normal rain, with a pH below 5.0. This phenomenon often results from the emission of sulfur dioxide (SO\(_2\)) and nitrogen oxides (NO\(_x\)) into the atmosphere. These gases can originate from industrial processes and transportation emissions.
  • The released pollutants react with water vapor present in the air to form sulfuric and nitric acids.
  • These acids lower the pH of rain, leading to more acidic conditions when it falls back to Earth's surface.
Acid rain can have harmful effects on plants, aquatic life, infrastructure, and even human health by altering the natural balance of the environment. It can damage forests, pollute water bodies, and erode buildings.
Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. In a normal distribution, it helps tell us how spread out the observations are around the mean.
  • The larger the standard deviation, the more spread out the values are.
  • A smaller standard deviation indicates that the values tend to be closer to the mean.
In our acid rain problem, the standard deviation is 0.54. This number tells us how much the pH values of rain typically vary from the average pH of 5.43. A normal distribution with a smaller standard deviation will gather its values more tightly around the mean, which is important to know when assessing the frequency of extreme pH values like those in acid rain scenarios.
Z-Score
A z-score is a statistical measure that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. Calculating a z-score helps us understand how unusual or usual a specific value is within a dataset.
  • The formula for a z-score is: \( z = \frac{X - \mu}{\sigma} \)
  • \( X \) is the value being evaluated, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
For the acid rain example, we found the z-score for a pH of 5.0 to be -0.796, indicating that this pH value lies \(-0.796\) standard deviations below the mean of 5.43. Z-scores allow us to use the standard normal distribution table to find out the proportion of values that fall below this level, which in this case tells us the percentage of the time rain is acidic, i.e., about 21.3% of the time.

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Most popular questions from this chapter

Cholesterol. Low-density lipoprotein, or LDL, is the main source of cholesterol buildup and blockage in the arteries. This is why LDL is known as "bad cholesterol." LDL is measured in milligrams per deciliter of blood, or mg/dL. In a population of adults at risk for cardiovascular problems, the distribution of LDL levels is Normal, with a mean of \(123 \mathrm{mg} / \mathrm{dL}\) and a standard deviation of 41 \(\mathrm{mg} / \mathrm{dL}\). If an individual's LDL is at least 1 standard deviation or more above the mean, he or she will be monitored carefully by a doctor. What percentage of individuals from this population will have LDL levels 1 or more standard deviations above the mean? Use the \(68-95-99.7\) rule.

Monsoon Rains. The summer monsoon rains in India follow approximately a Normal distribution with mean 852 millimeters \((\mathrm{mm})\) of rainfall and standard deviation \(82 \mathrm{~mm}\). (a) In the drought year \(1987,697 \mathrm{~mm}\) of rain fell. In what percent of all years will India have \(697 \mathrm{~mm}\) or less of monsoon rain? (b) "Normal rainfall" means within \(20 \%\) of the long-term average, or between \(682 \mathrm{~mm}\) and \(1022 \mathrm{~mm}\). In what percent of all years is the rainfall normal?

Upper Arm Lengths. The upper arm length of males over 20 years old in the United States is approximately Normal with mean \(39.1\) centimeters \((\mathrm{cm})\) and standard deviation \(2.3 \mathrm{~cm}\). Use the 68-95-99.7 rule to answer the following questions. (Start by making a sketch like Figure 3.10.) (a) What range of lengths covers the middle \(99.7 \%\) of this distribution? (b) What percent of men over 20 have upper arm lengths greater than \(41.4 \mathrm{~cm}\) ?

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 \(99 \%\)

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?
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