IQ Scores. In Exercises 5–8, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler IQ test).

검증된 단계별 안내

Step 1: Understand the problem. The graph represents a normal distribution of IQ scores with a mean (μ) of 100 and a standard deviation (σ) of 15. The shaded region corresponds to scores less than 118. We need to find the area of this shaded region, which represents the cumulative probability up to 118.

Step 2: Convert the IQ score of 118 into a z-score using the formula: z = (X - μ) / σ, where X is the IQ score, μ is the mean, and σ is the standard deviation. Substitute the values: X = 118, μ = 100, and σ = 15.

Step 3: Once the z-score is calculated, use a standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to this z-score. This cumulative probability represents the area under the curve to the left of the z-score.

Step 4: Interpret the cumulative probability. The value obtained from the z-table or software is the proportion of the population with IQ scores less than 118.

Step 5: If required, express the cumulative probability as a percentage by multiplying the value by 100. This percentage represents the likelihood of an individual having an IQ score less than 118.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

Z-Scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. For the IQ score of 118, the Z-score can be calculated to determine its position relative to the mean, which is essential for finding the area under the curve.

Area Under the Curve

The area under the curve in a normal distribution represents the probability of a score falling within a certain range. To find the area to the left of a specific score, such as 118, one can use Z-scores and standard normal distribution tables or software. This area corresponds to the proportion of the population that scores below that value.