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.

c. If 4 subjects take the Wechsler IQ test and they have a mean of 131 but the individual scores are lost, can we conclude that all 4 of them have scores of at least 131?

Verified step by step guidance

Step 1: Understand the problem. The question asks whether we can conclude that all 4 subjects have scores of at least 131, given that their mean score is 131. Recall that the mean is the average of the scores, and it does not necessarily imply that all individual scores are equal to or greater than the mean.

Step 2: Recall the formula for the mean: \( \text{Mean} = \frac{\text{Sum of all scores}}{\text{Number of scores}} \). Here, the mean is 131, and the number of scores is 4. Using this formula, calculate the total sum of the scores: \( \text{Sum of scores} = \text{Mean} \times \text{Number of scores} = 131 \times 4 \).

Step 3: Consider the possibility of variability in the individual scores. The mean of 131 only tells us the average, but individual scores could vary above or below the mean. For example, one score could be higher than 131 while another could be lower, as long as their average remains 131.

Step 4: To determine if all scores are at least 131, consider the extreme case where one or more scores are below 131. If even one score is below 131, then the others must compensate by being higher than 131 to maintain the mean. This means we cannot conclude definitively that all scores are at least 131 based solely on the mean.

Step 5: Conclude that without additional information about the individual scores or their distribution, it is not possible to determine whether all 4 subjects scored at least 131. The mean alone does not provide sufficient evidence to make this conclusion.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 interpreting how individual scores relate to the mean.

Mean and Individual Scores

The mean is the average of a set of values, calculated by summing all the scores and dividing by the number of scores. However, the mean does not provide information about the individual scores themselves. In this case, while the mean IQ score of the four subjects is 131, it does not guarantee that each individual score is at least 131, as some scores could be lower, provided they average out to 131.

Statistical Inference

Statistical inference involves drawing conclusions about a population based on a sample of data. In this scenario, even though the mean IQ score of the four subjects is known, we cannot infer that all individual scores meet a specific threshold without additional information. This highlights the importance of understanding the limitations of statistical conclusions when dealing with averages.

Watch next

Master Population Standard Deviation Known with a bite sized video explanation from Patrick

Start learning