(a) The probability that a randomly selected typist's speed is at most 60 wpm can be calculated by finding the area under the normal curve to the left of 60 wpm. This can be done by standardizing the value and using a z-table.

First, we calculate the z-score for 60 wpm:
z = (60 - 60) / 15 = 0

Next, we look up the probability for a z-score of 0 in the z-table. The probability corresponds to the area to the left of the z-score, which in this case is 0.5.

Therefore, the probability that a randomly selected typist's speed is at most 60 wpm is 0.5.

(b) The probability that a randomly selected typist's speed is between 30 and 75 wpm can be calculated by finding the area under the normal curve between these two values. Again, we need to standardize the values using z-scores.

For 30 wpm:
z1 = (30 - 60) / 15 = -2

For 75 wpm:
z2 = (75 - 60) / 15 = 1

Next, we look up the probabilities for these two z-scores in the z-table.

The probability for a z-score of -2 is 0.0228, and the probability for a z-score of 1 is 0.8413.

To find the probability between these two values, we subtract the probability for the lower z-score from the probability for the higher z-score:
0.8413 - 0.0228 = 0.8185

Therefore, the probability that a randomly selected typist's speed is between 30 and 75 wpm is 0.8185.

(c) The probability of finding a typist in this population whose speed exceeded 105 wpm can be calculated by finding the area to the right of 105 wpm under the normal curve.

First, we calculate the z-score for 105 wpm:
z = (105 - 60) / 15 = 3

Next, we look up the probability for a z-score of 3 in the z-table. The table gives us the probability to the left of the z-score. As we want the probability to the right of the z-score, we subtract the value from 1:
1 - 0.9987 = 0.0013

Therefore, the probability of finding a typist in this population whose speed exceeded 105 wpm is 0.0013.

(d) Since the speeds of two typists are independent, we can simply multiply the individual probabilities.

The probability that one typist's speed exceeds 90 wpm can be calculated as the area to the right of 90 wpm under the normal curve.

First, we calculate the z-score for 90 wpm:
z = (90 - 60) / 15 = 2

Next, we look up the probability for a z-score of 2 in the z-table. The table gives us the probability to the left of the z-score. As we want the probability to the right of the z-score, we subtract the value from 1:
1 - 0.9772 = 0.0228

Since the two typists are independent, the probability that both their speeds exceed 90 wpm is the product of the individual probabilities:
0.0228 * 0.0228 = 0.0005208

Therefore, the probability that both their speeds exceed 90 wpm is 0.0005208.

(e) To determine the typing speeds that qualify individuals for the special training, we need to find the cutoff value at which the slowest 15% of typists are included.

First, we need to find the z-score that corresponds to the 15th percentile by looking up the probability in the z-table.

The probability to the left of the z-score is 0.15. Looking up this probability in the z-table, we find that the z-score is approximately -1.04.

Next, we can use the z-score formula to find the typing speed at the 15th percentile:
z = (x - 60) / 15
-1.04 = (x - 60) / 15

Solving for x (typing speed), we have:
x - 60 = -1.04 * 15
x - 60 = -15.6
x = 44.4

Therefore, any typist with a typing speed of 44.4 wpm or less is in the slowest 15% of typists.

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