Final answer:
The probability questions associated with IQ score distributions can be resolved by calculating Z-scores and referencing the standard normal distribution table or using a statistical function on a calculator. For larger samples, the central limit theorem assists in probability estimations for sample mean scores. Specific percentiles can determine potential MENSA qualification or range for the middle 50 percent of IQ scores.
Step-by-step explanation:
Understanding IQ Score Distribution and Calculations
The intelligence quotient, or IQ, is a measure of a person's relative intelligence. IQ scores are distributed normally with a mean (μ) of 100 and a standard deviation (σ) of 15. When considering a random individual from this distribution, various probabilities can be calculated using the Z-score formula.
To calculate the probability that a person has an IQ greater than a certain value, we would find the area to the right of that value on the distribution curve. Conversely, for a score less than a specific value, we look for the area to the left. For example, the probability that someone has an IQ less than 125 can be found by converting the IQ score to a Z-score using the formula Z = (X - μ) / σ, where X is the IQ score. This Z-score is then referenced on a standard normal distribution table or calculated using a calculator equipped with statistics functions.
If we are examining a group of people, such as a sample of 600, we can determine how many might fall above or below a specific IQ level by multiplying the total number of people by the probability calculated for an individual.
Issues such as determining the probability that the sample mean scores will fall within a certain range for a group of people requires understanding of sampling distributions and potentially using the central limit theorem if the sample size is sufficiently large (usually n>30).
To identify individuals who may qualify for MENSA, which requires a score in the top 2 percent of all IQs, you would look for the IQ score associated with the 98th percentile of this distribution. This can be computed by finding the Z-score that corresponds to the 0.98 probability on the standard normal distribution and then translating this Z-score back into an IQ score.
The middle 50 percent of IQ scores, or the interquartile range (IQR), is identified by looking at the 25th percentile (Q1) and the 75th percentile (Q3) of the distribution. The Z-scores corresponding to these percentiles can be used to calculate the corresponding IQ scores.
For hypothesis testing about the mean IQ score, assuming the standard deviation is known and the sample size is sufficiently large, the normal distribution would typically be used rather than a student's t-distribution.