Mensa Membership in Mensa requires a score in the top 2% on a standard intelligence test. The Wechsler IQ test is designed for a mean of 100 and a standard deviation of 15, and scores are normally distributed.

a. Find the minimum Wechsler IQ test score that satisfies the Mensa requirement.

검증된 단계별 안내

Step 1: Understand the problem. Mensa membership requires a score in the top 2% of the population on a standard intelligence test. This means we need to find the IQ score corresponding to the 98th percentile of a normal distribution with a mean (μ) of 100 and a standard deviation (σ) of 15.

Step 2: Recall the properties of the normal distribution. The 98th percentile corresponds to a cumulative probability of 0.98. To find the z-score associated with this percentile, use a z-table or statistical software. The z-score is a standardized value that represents the number of standard deviations a data point is from the mean.

Step 3: Use the z-score formula to find the IQ score. The formula is:

x = μ + z \times σ

, where x is the IQ score, μ is the mean, z is the z-score, and σ is the standard deviation.

Step 4: Substitute the values into the formula. Use the mean (μ = 100), standard deviation (σ = 15), and the z-score obtained from the z-table or software for the 98th percentile.

Step 5: Solve the equation to find the minimum IQ score required for Mensa membership. This will give you the score corresponding to the top 2% of the population.

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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, indicating that data near the mean are more frequent in occurrence than data far from the mean. In the context of IQ scores, this means that most individuals score around the average (100), with fewer individuals scoring significantly higher or lower. Understanding this concept is crucial for determining percentiles and interpreting scores.

Z-Score

A Z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. In this case, to find the minimum IQ score for Mensa membership, we need to determine the Z-score that corresponds to the top 2% of the distribution, which helps in calculating the specific IQ score needed.

Percentiles

Percentiles are measures that indicate the relative standing of a value within a dataset. For example, being in the top 2% means that a score is higher than 98% of the scores in the distribution. To find the minimum IQ score for Mensa, we need to identify the score that corresponds to the 98th percentile of the Wechsler IQ test distribution, which is essential for meeting the membership criteria.

Mensa Membership in Mensa requires a score in the top 2% on a standard intelligence test. The Wechsler IQ test is designed for a mean of 100 and a standard deviation of 15, and scores are normally distributed.a. Find the minimum Wechsler IQ test score that satisfies the Mensa requirement.