Assume that adults have IQ scores that are normally distributed with a mean of 97.9 and a standard deviation of 18.2. Find the first quartile \( Q_1 \), which is the IQ score separating the bottom 25% from the top 75%.

(Hint: Draw a graph.)

The first quartile is __. (Type an integer or decimal rounded to one decimal place as needed.)

Question

18-03-2024
Mathematics

High School

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Assume that adults have IQ scores that are normally distributed with a mean of 97.9 and a standard deviation of 18.2. Find the first quartile \( Q_1 \), which is the IQ score separating the bottom 25% from the top 75%.

(Hint: Draw a graph.)

The first quartile is __. (Type an integer or decimal rounded to one decimal place as needed.)

Answer :

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noormalik3019 noormalik3019 · Answer 1 · 2023-08-31T13:06:06-04:00

Final answer: The first quartile is 85.4.

Explanation: The first quartile, denoted as Q₁, is the value below which 25% of the data falls. To find Q₁, we need to determine the z-score corresponding to the 25th percentile. Using the standard normal distribution table, we find the z-score for the 25th percentile is approximately -0.6745. Next, we use the z-score formula: z = (X - μ) / σ, where X is the raw score, μ is the mean, and σ is the standard deviation. Rearranging the formula to solve for X, we get X = μ + (z * σ). Substituting the given values (mean = 97.9, standard deviation = 18.2, and z = -0.6745) into the formula, we find the first quartile IQ score: 85.4.

In summary, the first quartile IQ score of 85.4 is obtained by finding the z-score corresponding to the 25th percentile and using it to calculate the raw score based on the given mean and standard deviation. This process ensures an accurate placement of the first quartile, separating the bottom 25% from the top 75% of the normally distributed adult IQ scores.