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Normal Distribution IQ Score Problem: Find Percentage of 20-34 Age Group with IQ Below 100
Statistics
High School Grade 11/12
Question Content
Scores on the Wechsler Adult Intelligence Scale, a standard IQ test, are approximately Normal for the age group of 20 to 34. The mean score is 110 with a standard deviation of 25. What percent of this age group has an IQ below 100? a) Write the inequality using the raw score. b) Find the z-score and rewrite the inequality. c) Plot this z-score on the standard normal curve. d) Find and interpret the percentage.
Correct Answer
a) $X < 100$; b) $z = -0.4$, $Z < -0.4$; c) Plot a point at -0.4 on the horizontal axis of the standard normal curve (centered at 0) and shade the area to the left; d) Approximately 34.46% of the 20-34 age group has an IQ below 100.
Detailed Solution Steps
1
Step 1: For part a, define $X$ as the raw IQ score. The question asks for scores below 100, so the inequality is $X < 100$.
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Step 2: For part b, use the z-score formula $z = \\frac{X - \\mu}{\\sigma}$, where $\\mu = 110$ (mean) and $\\sigma = 25$ (standard deviation). Substitute $X=100$: $z = \\frac{100 - 110}{25} = -0.4$. The rewritten inequality for the standard normal variable $Z$ is $Z < -0.4$.
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Step 3: For part c, draw a standard normal curve (bell-shaped, symmetric about 0). Mark -0.4 on the left side of the center (0) on the horizontal axis, then shade the entire area under the curve to the left of -0.4.
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Step 4: For part d, use a standard normal table or calculator to find the area to the left of $z=-0.4$. This area is approximately 0.3446, or 34.46%. Interpret this as: about 34.46% of people aged 20-34 have an IQ score lower than 100.
Knowledge Points Involved
1
Raw Score vs. Z-Score
A raw score is the original observed value (like an IQ score). A z-score transforms a raw score to a standard scale using $z = \\frac{X - \\mu}{\\sigma}$, where $\\mu$ is the population mean and $\\sigma$ is the population standard deviation. It measures how many standard deviations a value is from the mean, used to compare values across different normal distributions.
2
Standard Normal Distribution
A special normal distribution with a mean of 0 and standard deviation of 1. It is symmetric, bell-shaped, and the total area under its curve is 1 (or 100%). It is used to find probabilities/percentages for any normal distribution by converting raw scores to z-scores.
3
Normal Distribution Probability Calculation
To find the percentage of data in a range for a normal distribution, convert raw scores to z-scores, then use a standard normal table, calculator, or statistical software to find the corresponding area under the standard normal curve. The area represents the probability/percent of data in that range.
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