Answer :
Final answer:
To calculate the probability of picking one person with an IQ between 78 and 94, we convert the IQ scores to z-scores and use them to find the corresponding areas under the standard normal distribution curve. The final probability is the difference between these two areas.
Explanation:
The question is asking for the probability of picking one person from a given population who has an IQ between 78 and 94, given that the average IQ is 100 and the standard deviation is 12. To find this probability, we need to use the standard normal distribution, as IQ scores are typically normally distributed. We'll calculate the z-scores for 78 and 94, and then use a z-table or a statistical software to find the probabilities corresponding to these z-scores.
First, find the z-score for 78: z = (78 - 100) / 12 = -22 / 12 = -1.83. Now, find the z-score for 94: z = (94 - 100) / 12 = -6 / 12 = -0.5. After finding the z-scores, look up the area to the left of each score on the z-table and subtract the smaller area from the larger one to get the probability that an individual's IQ is between 78 and 94.
This calculation gives you the probability of selecting a person whose IQ score is greater than 78 but less than 94 from the population. In general terms, a z-score provides a way to compare individual scores to the average of a dataset, measuring the number of standard deviations a data point is from the mean. In this context, it allows us to find the desired probability.
