Answer :
Final answer:
To find the first quartile (Q1) of IQ scores, use the formula z = (X - mean) / standard deviation to calculate the z-score. Then, find the corresponding z-score for the bottom 25% using a table or calculator with the inverse normal distribution function. Solve for Q1 using the formula Q1 = (z * standard deviation) + mean. The first quartile is approximately 85.44 for the given IQ scores.
Explanation:
To find the first quartile (Q1), which separates the bottom 25% from the top 75% of IQ scores, we need to first find the corresponding z-score. The formula to calculate the z-score is:
z = (X - mean) / standard deviation
Plugging in the values given in the question:
z = (Q1 - 100.5) / 22.6
We can use the z-score table or a calculator with the inverse normal distribution function to find the corresponding z-score for the bottom 25%, which is approximately -0.674 for Q1. Now, we can solve for Q1:
Q1 = (z * standard deviation) + mean
Q1 = (-0.674 * 22.6) + 100.5
Q1 = 85.44
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Final answer:
The first quartile (Q1) of an IQ score distribution can be found by using the inverse norm function, considering 0.25 as the percentile, 100.5 as the mean and 22.6 as the standard deviation. This Q1 value is the IQ score below which 25% of the adult population can be found.
Explanation:
The question is asking for the first quartile (Q1) of adult IQ scores, which are normally distributed with a mean of 100.5 and a standard deviation of 22.6. The first quartile is essentially the 25th percentile of the distribution, meaning the value below which 25% of the data points can be found.
To calculate this, we would normally use a z-table or a statistical function like 'inverse norm' in a calculator that calculates z-scores. The formula would be =invNorm(0.25, 100.5, 22.6). However, without a calculator or a z-table at hand, I am unable to provide the exact numeric value for Q1 at this time.
In general, remember that quartiles are measures that help us to understand the distribution of data in the set. The first quartile splits off the lowest 25% of data from the highest 75%.
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