Question

problem 4
scores on an iq test are normally distributed with a mean of 100 and a standard deviation of 15. use technology to find the iq scores that mark the boundaries for each group.

1. what iq score marks the 80th percentile (the score that 80% of test-takers score at or below)?
2. what iq score marks the 10th percentile (the score that 10% of test-takers score at or below)?
3. what two iq scores mark the boundaries of the middle 60% of test-takers? (hint: find the 20th and 80th percentiles.)

Explanation:

Step1: Recall normal distribution formula

The formula for a value \(X\) from a normal distribution is:
$$X = \mu + z \cdot \sigma$$
where \(\mu = 100\) (mean), \(\sigma = 15\) (standard deviation), and \(z\) is the z-score corresponding to the percentile.

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For Question 1 (80th percentile):

Step2: Find 80th percentile z-score

Using technology, the z-score for the 80th percentile is \(z \approx 0.84\).

Step3: Calculate IQ score

$$X = 100 + 0.84 \cdot 15$$
$$X = 100 + 12.6 = 112.6$$

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For Question 2 (10th percentile):

Step4: Find 10th percentile z-score

Using technology, the z-score for the 10th percentile is \(z \approx -1.28\).

Step5: Calculate IQ score

$$X = 100 + (-1.28) \cdot 15$$
$$X = 100 - 19.2 = 80.8$$

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For Question 3 (middle 60%: 20th and 80th percentiles):

Step6: Find 20th percentile z-score

Using technology, the z-score for the 20th percentile is \(z \approx -0.84\).

Step7: Calculate 20th percentile IQ score

$$X = 100 + (-0.84) \cdot 15$$
$$X = 100 - 12.6 = 87.4$$

Step8: Use 80th percentile result

From Step3, the 80th percentile IQ score is \(112.6\).

Answer:

1. 112.6
2. 80.8
3. 87.4 and 112.6