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Chapter 5: Problem 26

Some IQ tests are standardized to a Normal model, with a mean of 100 and a standard deviation of 16 a) Draw the model for these IQ scores. Clearly label it, showing what the \(68-95-99.7\) Rule predicts. b) In what interval would you expect the central \(95 \%\) of IQ scores to be found? c) About what percent of people should have IQ scores above \(116 ?\) d) About what percent of people should have IQ scores between 68 and \(84 ?\) c) About what percent of people should have IQ scores above \(132 ?\)

Short Answer

Expert verified
a) Model has mean 100, std. dev. 16, 68-95-99.7 rule shows intervals. b) 68 to 132. c) About 16%. d) About 13.5%. e) About 5%.

Step by step solution

01

Describe and sketch the Normal distribution

IQ scores follow a Normal distribution, where the mean (average) is 100 and the standard deviation is 16. The Normal distribution is symmetric and bell-shaped. According to the 68-95-99.7 Rule, approximately 68% of the data lie within one standard deviation of the mean (84 to 116), 95% lie within two standard deviations (68 to 132), and 99.7% lie within three standard deviations (52 to 148).
02

Find the central 95% interval

Using the 68-95-99.7 Rule, the central 95% of the data is within two standard deviations of the mean. Since the mean is 100 and the standard deviation is 16, two standard deviations add or subtract 32 from the mean: \[ 100 - 2(16) = 68 \quad\text{and}\quad 100 + 2(16) = 132. \]Thus, the central 95% of IQ scores will be between 68 and 132.
03

Calculate percent above an IQ of 116

To find the percentage of people with an IQ above 116, note that 116 is one standard deviation above the mean. Using the 68-95-99.7 Rule, 34% of the population is between the mean (100) and one standard deviation above (116). Thus, 50% (total above mean) - 34% = 16% of people have an IQ above 116.
04

Calculate percent between 68 and 84

The standard deviations for IQ 68 and 84 are -2 and -1, respectively. The percentage of people between -2 and -1 standard deviations (68 and 84) is half of the 34% between -1 and 1 standard deviations. Therefore, 13.5% of the population has an IQ between 68 and 84.
05

Calculate percent above an IQ of 132

To find the percentage of people with an IQ above 132, recognize that 132 is two standard deviations above the mean. According to the 68-95-99.7 Rule, 95% of the population is below two standard deviations (i.e., below 132). Hence, 100% - 95% = 5% of people should have an IQ above 132.

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Key Concepts

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

68-95-99.7 Rule
The 68-95-99.7 Rule, often referred to as the Empirical Rule, is a fundamental concept in understanding the characteristics of the normal distribution. It provides a clear guide of how data is spread when it follows a normal distribution. Let鈥檚 break it down:
  • 68%: This percentage tells us that about 68% of the data lies within one standard deviation from the mean. So, if you鈥檙e looking at test scores or heights that follow a normal model, the majority of them are pretty close to average.
  • 95%: This percentage extends the coverage to two standard deviations from the mean. As we apply this to the exercise, for IQ scores with a mean of 100 and a standard deviation of 16, we find that 95% of people have IQ scores between 68 and 132.
  • 99.7%: Finally, almost all data, specifically 99.7%, falls within three standard deviations from the mean. These boundaries, though far out, capture nearly the entire data set.
This rule is particularly handy in predicting the spread and likelihood of various outcomes in a simple manner. It forms the backbone of understanding variability in normal distributions and is crucial for interpreting statistical data.
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are. In simpler terms, it tells us "how much stuff varies" or "deviates" from the average (mean). The larger the standard deviation, the more spread out the values. To see this in action, consider the IQ example with a standard deviation of 16:
  • A small standard deviation indicates that the values tend to be close to the mean. In our IQ example, scores near 100 are very common, and you won't find too many scores that wander too far away.
  • A large standard deviation, on the other hand, would mean IQ scores are much more spread out from the mean of 100.
Standard deviation plays a pivotal role in the 68-95-99.7 Rule by determining how wide the intervals will be. It's a cornerstone of statistics, essential for understanding data variability and predicting patterns.
Mean
The mean is a fundamental concept in statistics. Simply put, it鈥檚 the "average" of a group of numbers. You find it by adding together all the numbers in a set and then dividing by the total count of numbers. In the IQ score example, the mean is 100. This number helps us understand what is typical or expected when looking at a set of IQ scores. It acts as a central point for the data distribution.
  • The mean is key for determining where the middle of a normal distribution lies.
  • It provides a reference point for evaluating how scores, like 116 or 132, compare to an average score.
Understanding the mean is crucial because it directly influences the discussion of standard deviation and how the normal distribution is formed. Without the mean, we wouldn鈥檛 have a baseline to measure the spread and variety within our data.

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Most popular questions from this chapter

The winning scores of all college men's basketball games for the \(2011-12\) season were approximately normally distributed with mean 77.5 points and standard deviation 12.5 points. a) Draw the Normal model for winning scores. b) What interval of winning scores would be the central \(95 \%\) of all winning scores for the \(2011-12\) season? c) About what percent of the winning scores should be less than 65 points? d) About what percent of the winning scores should be between 65 and 102 points? e) About what percent of the winning scores should be over 102 points?

Cars currently sold in the United States have an average of 135 horsepower, with a standard deviation of 40 horsepower. What's the z-score for a car with 195 horsepower?

An incoming freshman took her college's placement exams in French and mathematics. In French, she scored 82 and in math \(86 .\) The overall results on the French exam had a mean of 72 and a standard deviation of \(8,\) while the mean math score was 68 with a standard deviation of \(12 .\) On which exam did she do better compared with the other freshmen?

Recall that the beef cattle described in Exercise 17 had a mean weight of 1152 pounds, with a standard deviation of 84 pounds. a) Cattle buyers hope that yearling Angus steers will weigh at least 1000 pounds. To see how much over (or under) that goal the cattle are, we could subtract 1000 pounds from all the weights. What would the new mean and standard deviation be? b) Suppose such cattle sell at auction for 40 cents a pound. Find the mean and standard deviation of the sale prices for all the steers.

The first Stat exam had a mean of 80 and a standard deviation of 4 points; the second had a mean of 70 and a standard deviation of 15 points. Reginald scored an 80 on the first test and an 85 on the second. Sara scored an 88 on the first but only a 65 on the second. Although Reginald's total score is higher, Sara feels she should get the higher grade. Explain her point of view.
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