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Chapter 3: Problem 29

The Empirical Rule The Stanford-Binet Intelligence Quotient (IQ) measures intelligence. IQ scores have a bellshaped distribution with a mean of 100 and a standard deviation of \(15 .\) (a) What percentage of people has an IQ score between 70 and \(130 ?\) (b) What percentage of people has an IQ score less than 70 or greater than \(130 ?\) (c) What percentage of people has an IQ score greater than \(130 ?\)

Short Answer

Expert verified
(a) 95% (b) 5% (c) 2.5%

Step by step solution

01

Understand the Empirical Rule

The Empirical Rule states that for a bell-shaped distribution: 1. About 68% of the data lies within 1 standard deviation of the mean. 2. About 95% lies within 2 standard deviations. 3. About 99.7% lies within 3 standard deviations.
02

Calculate 1 Standard Deviation from the Mean

The mean IQ score is 100 and the standard deviation is 15. Therefore, 1 standard deviation from the mean is: Mean - 1SD = 100 - 15 = 85 Mean + 1SD = 100 + 15 = 115
03

Calculate 2 Standard Deviations from the Mean

For 2 standard deviations from the mean: Mean - 2SD = 100 - 30 = 70 Mean + 2SD = 100 + 30 = 130
04

Answer Percentage for IQ between 70 and 130

According to the Empirical Rule, about 95% of the data falls within 2 standard deviations from the mean. Therefore, 95% of people have an IQ between 70 and 130.
05

Answer Percentage for IQ less than 70 or greater than 130

Since 95% of people have IQ scores between 70 and 130, the percentage of people with an IQ less than 70 or greater than 130 is: 100% - 95% = 5%
06

Answer Percentage for IQ greater than 130

The Empirical Rule states that 2.5% of the population lies above 2 standard deviations from the mean (100 + 2 * 15). Thus, 2.5% of people have an IQ greater than 130.

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Key Concepts

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

IQ distribution
IQ scores follow a specific trend called a 'distribution'. A well-known type is the bell-shaped distribution (also called normal distribution). Imagine drawing all IQ scores on a graph; it would create a shape like a bell. The middle of the bell has the highest point, representing the average IQ of 100.
This means most people have IQ scores around 100. The curve gets lower as you move away from 100. Farther from the center, you鈥檒l see fewer people with those IQ scores. Hence, most people's IQs cluster around the average, creating the 'bell' shape.
standard deviation
Standard deviation shows how spread out numbers are from the average. In the IQ example, the average (mean) IQ score is 100, and the standard deviation is 15.
This means that most people's IQs are within 15 points above or below 100. So, IQ scores between 85 and 115 are common. Not coincidentally, this matches the first standard deviation in a bell-shaped curve, where most data points lie.
If you move further away, say 30 points above or below 100, that's the second standard deviation. The formula for it would be \[100 \text{ (mean)} \times 2 \text{ (standard deviation)} = \begin{cases} 70 \text{ (low end)} \ 130 \text{ (high end)} \text{.}\begin{cases}\]
percentage calculation
When dealing with bell-shaped distributions, like the IQ scores, there's a useful tool called the Empirical Rule to estimate percentages.
This rule tells us how much data falls within certain standard deviations from the mean:
  • About 68% of the data: falls within 1 standard deviation (between 85 and 115 for IQ)
  • About 95% falls within 2 standard deviations (between 70 and 130 for IQ)
  • About 99.7% falls within 3 standard deviations

For example, if you're asked what percentage of people have an IQ between 70 and 130, the answer, according to the Empirical Rule, is 95%.
bell-shaped distribution
The bell-shaped distribution is crucial in statistics to understand how a trait, like IQ, spreads out in a population.
The characteristics of this curve make it predictable and symmetric around the mean:
  • Most people score near the average (middle of the bell)
  • As you move away from the average, fewer people have those scores
A typical bell-shaped curve is dense in the center and thin at the tails.
This distribution allows for precise statistical calculations, making it a powerful tool in assessing and predicting probabilities and percentages in any given dataset.

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Most popular questions from this chapter

The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic; a pH greater than 7 is alkaline. The following data represent the \(\mathrm{pH}\) in samples of bottled water and tap water. $$ \begin{array}{lllllll} \hline \text { Tap } & 7.64 & 7.45 & 7.47 & 7.50 & 7.68 & 7.69 \\ & 7.45 & 7.10 & 7.56 & 7.47 & 7.52 & 7.47 \\ \hline \text { Bottled } & 5.15 & 5.09 & 5.26 & 5.20 & 5.02 & 5.23 \\ & 5.28 & 5.26 & 5.13 & 5.26 & 5.21 & 5.24 \\ \hline \end{array} $$ (a) Determine the mean, median, and mode \(\mathrm{pH}\) for each type of water. Comment on the differences between the two water types. (b) Suppose the \(\mathrm{pH}\) of 7.10 in tap water was incorrectly recorded as \(1.70 .\) How does this affect the mean? the median? What property of the median does this illustrate?

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The data below represent the age of the mother at the time of her first birth for a random sample of 30 mothers. $$ \begin{array}{llllll} \hline 21 & 35 & 33 & 25 & 22 & 26 \\ \hline 21 & 24 & 16 & 32 & 25 & 20 \\ \hline 30 & 20 & 20 & 29 & 21 & 19 \\ \hline 18 & 24 & 33 & 22 & 23 & 25 \\ \hline 17 & 23 & 25 & 29 & 25 & 19 \\ \hline \end{array} $$ (a) Construct a box plot of the data. (b) Use the box plot and quartiles to describe the shape of the distribution.
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