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QuestionCompare IQs using z-scores: Test A (mean 100, SD 14) score 127 vs Test B (mean 100, SD 16) score 130. Who is higher?

Studdy Solution

STEP 1

Assumptions1. IQ scores on Test A and Test B are normally distributed. . The mean IQ score for Test A is100 with a standard deviation of14.
3. The mean IQ score for Test B is100 with a standard deviation of16.
4. Individual1 scored127 on Test A.
5. Individual scored130 on Test B.

STEP 2

We need to calculate the z-score for each individual's IQ score. The z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formulaz=x−μσz = \frac{x - \mu}{\sigma}where xx is the element, μ\mu is the mean, and σ\sigma is the standard deviation.

STEP 3

First, let's calculate the z-score for Individual1 who scored127 on Test A. Plug in the values for xx, μ\mu, and σ\sigma into the z-score formula.
zA=127−10014z_A = \frac{127 -100}{14}

STEP 4

Calculate the z-score for Individual1.
zA=127−10014=1.93z_A = \frac{127 -100}{14} =1.93

STEP 5

Next, let's calculate the z-score for Individual2 who scored130 on Test B. Plug in the values for xx, μ\mu, and σ\sigma into the z-score formula.
zB=130−10016z_B = \frac{130 -100}{16}

STEP 6

Calculate the z-score for Individual2.
zB=130−10016=1.875z_B = \frac{130 -100}{16} =1.875

STEP 7

Now that we have the z-scores for both individuals, we can compare them to determine who has the higher IQ. Since a higher z-score indicates a score further above the mean, the individual with the higher z-score has the higher IQ.
The individual who scores127 on Test A has a higher IQ because1.93 >1.875.
So, the answer is B. The individual who scores127 on Test A.

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