Scores on the Wechsler Adult Intelligence Scale, a standard IQ test, are approximately normal for the 20 to 34 age group with a mean [tex]y = 110[/tex] and standard deviation [tex]o = 25[/tex].

a. What percent of this age group have an IQ less than 100?

b. What percent of this age group have an IQ between 90 and 115?

c. Find the 8th percentile of the IQ scores distribution of 20 to 34 year olds.

d. Find the IQ score which separates the lowest 25% of all IQ scores for this age group from the highest 75%.

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On average, a certain computer part lasts 12 years. The length of time the computer part lasts is exponentially distributed.

a. What is the probability that a computer part lasts more than 7 years?

b. On average, how long would 5 computer parts last if they are used one after another?

c. Eighty percent of computer parts last at most how long?

d. What is the probability that a computer part lasts between 9 and 11 years?

a. To find the percent of the age group with an IQ less than 100, we can use the standard normal distribution and the given mean [tex](\(\mu = 110\))[/tex]and standard deviation [tex](\(\sigma = 25\))[/tex]. We need to find the z-score for[tex]\(x = 100\)[/tex] and then find the area to the left of that z-score using a standard normal table or calculator.

b. To find the percent of the age group with an IQ between 90 and 115, we can find the z-scores for [tex]\(x = 90\) and \(x = 115\)[/tex] and then find the area between those two z-scores using the standard normal distribution.

c. To find the 8th percentile of the IQ scores distribution, we need to find the z-score corresponding to the 8th percentile and then use the z-score formula to find the corresponding IQ score.

d. To find the IQ score that separates the lowest 25% from the highest 75%, we need to find the z-score that corresponds to the 25th percentile and then use the z-score formula to find the corresponding IQ score.

For the computer part problem, similar steps can be followed for each part (a, b, c, d) using the exponential distribution formula and the given average[tex](\(\lambda = \frac{1}{12}\)).[/tex]

The answers to these questions involve using probability distributions and their properties. For the normal distribution problems, we utilize z-scores to translate the given values into the standard normal distribution, which has a mean of 0 and a standard deviation of 1. By finding the area under the standard normal curve corresponding to specific z-scores, we can determine the probabilities or percentiles required.

Similarly, for the exponential distribution, we use the properties of the exponential distribution to find probabilities and expected values. These calculations provide insights into the characteristics of the given data and help answer questions about the distribution of IQ scores and computer part lifetimes in the specified age group.

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