Question

IQ Tests A standard intelligence test has an average score of \(100 .\) According to statistical theory, of the people who take the test, the \(2.5 \%\) with the highest scores will have scores of more than \(1.96 \sigma\) above the average, where \(\sigma\) (sigma, a number called the standard deviation) depends on the nature of the test. If \(\sigma=12\) for this test and there is (in principle) no upper limit to the score possible on the test, write the interval of possible test scores of the people in the top \(2.5 \%\)

Short answer

The interval is \([123.52, \infty)\).

Step-by-step solution

Identify the given values

First, note the provided values: the average score is given as 100, the standard deviation (\(\sigma\)) is 12, and the top 2.5% of scores are those that are more than 1.96\(\sigma\) above the average.

Calculate the cutoff score

To find the score that separates the top 2.5% from the rest, use the z-score formula: \(Z = \frac{(X - \mu)}{\sigma}\). Here, \(Z = 1.96\) (since we're looking for the top 2.5%), \(\mu\) (the mean) = 100, and \(\sigma\) = 12. Solve for \(X\) (the score): \(1.96 = \frac{(X - 100)}{12}\).

Solve for the score X

Rearrange the equation to solve for \(X\): \(X = 1.96 \times 12 + 100\).

Compute the final value

Calculate the expression: \(1.96 \times 12 = 23.52\) and then add this to the mean: \(23.52 + 100 = 123.52\).

Write the interval

Since there is no upper limit to the scores, the interval for the top 2.5% is \([123.52, \infty)\).

Key concepts

Standard Deviation

Standard deviation (\(\sigma\)) measures how spread out the values in a data set are. It's a crucial concept in statistics, helping us understand the variability within a set of data.
A low standard deviation means that the values tend to be close to the mean (average) value, whereas a high standard deviation means the values are spread out over a wider range.
In our exercise, the standard deviation is given as 12. This value tells us how much the IQ test scores typically deviate from the average score of 100. If most people's scores are close to this average, the standard deviation helps quantify that closeness or divergence.
Understanding standard deviation allows us to make sense of the distribution of scores and predict the likelihood of certain values appearing. In practical terms, it can hint at whether a data point is common or unusual in its context.

Z-Score

A z-score helps us understand how far away a particular value is from the mean, measured in units of the standard deviation. The formula for calculating a z-score is: \(\[ Z = \frac{(X - \mu)}{\sigma} \]\), where \(X\) is the value in question, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
In our example, we are given that the top 2.5% of scores have a z-score of +1.96. This means these scores are 1.96 standard deviations above the mean. By using this z-score, we can find the exact score that marks the cutoff for the top 2.5% of IQ test takers.
We use the z-score formula to calculate this cutoff: \(\[ 1.96 = \frac{(X - 100)}{12} \]\). By solving this equation, we find the value of \(X\), which represents the minimum score required to be in the top 2.5%.

Percentile

Percentiles indicate the relative standing of a value within a data set. If you score in the 90th percentile, this means you scored better than 90% of the people.
In the context of our IQ test example, being in the top 2.5% means you are above the 97.5th percentile, since 100% - 2.5% = 97.5%.
We determine this by calculating the z-score that corresponds to the desired percentile. For the top 2.5%, we use a z-score of +1.96. This z-score tells us how many standard deviations above the mean the cutoff score is. Using this concept, we can pinpoint specific values within any set of data to see where they fall relative to the rest of the dataset.

Mean

The mean (\(\mu\)), or average, is a measure of central tendency, which helps us understand the center of a data set. It's calculated by summing all the values in the dataset and then dividing by the number of values.
In our exercise, the mean IQ score is given as 100. This is the central value around which all other scores are distributed. Understanding the mean is essential because it provides a reference point for calculating other statistics, such as standard deviation and z-scores.
For example, when calculating the z-score for the top 2.5% of IQ scores, we used the mean of 100 in our formula: \(\[ Z = \frac{(X - 100)}{12} \]\). This helps to determine how much higher or lower a particular score is compared to the average score. Knowing the mean allows us to position individual scores within the whole dataset accurately.