Answer :
Final answer:
The first quartile (Q1) of adults' IQ scores can be found using the z-score formula. By calculating the z-score that corresponds to the cumulative probability of 0.25, we can determine the IQ score that separates the bottom 25%. The first quartile is approximately 86.49.
Explanation:
To find the first quartile, we need to determine the IQ score separating the bottom 25% of adults. Since the IQ scores are normally distributed, we can use the z-score formula to find the corresponding value. The first quartile (Q1) corresponds to the z-score that leaves 25% of the area under the curve to the left.
To find the z-score, we use the formula:
z = (X - mean) / standard deviation
Substituting the given values:
z = (X - 97.9) / 15.2
Since we want to find the IQ score that corresponds to the bottom 25%, we need to find the z-score that corresponds to the cumulative probability of 0.25. Using a standard normal distribution table, we find that the z-score is approximately -0.6745.
Now, we can solve for X:
-0.6745 = (X - 97.9) / 15.2
X - 97.9 = -0.6745 * 15.2
X = -0.6745 * 15.2 + 97.9
X ≈ 86.49
Therefore, the first quartile (Q1) is approximately 86.49. This means that 25% of adults have an IQ score of 86.49 or less.
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