Question

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 ? e) About what percent of people should have IQ scores above \(132 ?\)

Short answer

a) Use the normal curve with mean 100 and mark \( \,1\sigma, \,2\sigma, \,3\sigma\) intervals. b) 68 to 132. c) About 15.87%. d) About 18.26%. e) About 2.28%.

Step-by-step solution

Understand the Normal Distribution

The given IQ scores follow a normal distribution model with a mean \( \mu \) of 100 and a standard deviation \( \sigma \) of 16. The normal distribution is symmetric around the mean.

Visual Representation of the Model

Draw a bell-shaped normal curve centered at 100. Use the \(68-95-99.7\) rule to mark standard deviations: 68% of data falls within \( \mu \pm \sigma \), 95% within \( \mu \pm 2\sigma \), and 99.7% within \( \mu \pm 3\sigma \).

Calculating the 95% Interval

The 95% interval is within \(2\sigma\) of the mean. Calculate the interval as \(100 - 2(16) \) to \(100 + 2(16)\), which is 68 to 132.

Percent of People with IQ above 116

Convert the IQ score 116 into the z-score using \( z = \frac{116 - 100}{16} = 1\). From the standard normal distribution table, \( P(Z > 1) \approx 0.1587 \) or 15.87%.

Percent of People with IQ between 68 and 84

Convert both IQ scores into z-scores: for 68, \( z = \frac{68 - 100}{16} = -2\) and for 84, \( z = \frac{84 - 100}{16} = -1\). From the standard normal distribution, \( P(-2 < Z < -1) \approx 0.3413 - 0.1587 = 0.1826 \) or 18.26%.

Percent of People with IQ above 132

Convert the IQ score 132 into the z-score: \( z = \frac{132 - 100}{16} = 2\). The probability \( P(Z > 2) \approx 0.0228 \) or 2.28%.

Key concepts

Standard Deviation

The standard deviation is a measure that tells us how much the numbers in a data set typically differ from the average (mean) of the set. In the context of a normal distribution, it plays a key role in understanding the spread of the distribution.
In our IQ test example, the standard deviation is 16. This means that most IQ scores tend to hover within 16 points of the average IQ score of 100. Having a smaller standard deviation would mean that the IQ scores are closer to the mean, reflecting lesser individual variation, whereas a larger value would mean greater variety in IQ scores.

• It's used to calculate ranges of values where different portions of the data are expected.
• It helps determine how uncommon a particular score is relative to others.

In normal distributions, key intervals are measured using standard deviations from the mean, helping us to quickly see where most of the data lies and make predictions.

Z-Score

A z-score tells us how many standard deviations away a value is from the mean. It converts a particular data point into a standardized form, providing a clear sense of its relative position in the distribution.
To calculate a z-score, subtract the mean from the point and divide the result by the standard deviation. For example, an IQ score of 116 translates to a z-score:\[z = \frac{116 - 100}{16} = 1\]This calculation shows that 116 is one standard deviation above the mean.

• Z-scores help determine probabilities for values in a normal distribution.
• They allow comparison of data points across different distributions by standardizing them.

Using z-scores, we can look up corresponding probabilities in a standard normal distribution table, or use statistical software to find the proportion of data above, below, or between certain values.

68-95-99.7 Rule

The 68-95-99.7 rule, also known as the empirical rule, helps us understand the distribution of data when plotted on a normal curve. It gives us a quick overview of where the majority of data lies relative to the mean.
According to the rule:

• About 68% of values fall within one standard deviation from the mean.
• Roughly 95% are within two standard deviations.
• And nearly 99.7% are within three standard deviations.

Applying this to our IQ distribution, we see that: - 68% of IQ scores fall between 84 (100 - 16) and 116 (100 + 16). - 95% range from 68 (100 - 2×16) to 132 (100 + 2×16). - 99.7% lie between 52 (100 - 3×16) and 148 (100 + 3×16). This rule is handy for quickly estimating which values are common or rare in any normal distribution.