Answer :
Final answer:
The first quartile Q1 of the normally distributed IQ scores, which is the 25th percentile, is found by converting the percentile to a z-score and then back to an IQ score. The calculated IQ score for the first quartile is approximately (A)83.6.
Explanation:
To find the first quartile (Q1) of a normally distributed set of IQ scores with a mean of 97.3 and a standard deviation of 15.9, we need to determine the IQ score that marks the 25th percentile. The first quartile Q1 separates the lowest 25% of the distribution from the rest. To do this, we can use the standard normal distribution (z-table) after converting the IQ score to a z-score using the formula:
Z = (X - [tex]\(\mu\\[/tex]) / [tex]\\(\sigma\\[/tex])
Where Z is the z-score, X is the IQ score, \\(mu\\) is the mean, and \\(sigma\\) is the standard deviation. To find Q1, we look for a z-score that corresponds to the cumulative probability of 0.25 (25%). Looking up the z-score for 0.25 in the standard normal distribution table, we find that Z \\approx -0.675. Next, we convert Z back to the IQ score:
X = Z * [tex]\\(\sigma\\[/tex]) + [tex]\(\mu\\[/tex])
Substituting the values we get:
X = -0.675 * 15.9 + 97.3 = 83.5735
The nearest listed option to this calculated value is 83.6, which is choice (a).
