Question

Suppose that the distribution of typing speed in words per minute (wpm) for experienced typists using a new type of split keyboard can be approximated by a normal curve with mean 60 wpm and standard deviation 15 wpm ("The Effects of Split Keyboard Geometry on Upper Body Postures, Ergonomics [2009]: 104-111). a. What is the probability that a randomly selected typist's speed is at most 60 wpm? Less than 60 wpm? b. What is the probability that a randomly selected typist's speed is between 45 and 90 wpm? c. Would you be surprised to find a typist in this population whose speed exceeded 105 wpm? d. Suppose that two typists are independently selected. What is the probability that both their speeds exceed 75 wpm? e. Suppose that special training is to be made available to the slowest \(20 \%\) of the typists. What typing speeds would qualify individuals for this training? (Hint: See Example 6.23\()\)

Short answer

a. The probability is 50%. b. The probability is 81.85%. c. Yes, because only about 0.13% of typists would type faster than 105 wpm. d. The probability is approximately 2.52%. e. Typists who type at speeds slower than approximately 47.4 wpm would qualify for the training.

Step-by-step solution

Calculate probability for a randomly selected typist's speed being at most 60 wpm

To answer this, we need to calculate a z score which helps us know how many standard deviations an element is from the mean. Here, the speed is the same as the mean hence z score = \((60-60) / 15 = 0\). Under the standard normal distribution curve, a z-score of 0 corresponds to a cumulative probability of 0.5 or 50%.

Calculate the probability for speed being less than 60 wpm

For values less than the mean in a normal distribution, the probability is also 0.5 or 50%.

Calculate the probability for speed being between 45 and 90 wpm

To calculate this range we need two z scores - one for 45 wpm and one for 90 wpm. Z1 = \((45 - 60) / 15 = -1\) and Z2 = \((90 - 60) / 15 = 2\). Looking up these z values in the standard normal table, we find that P(Z < -1) = 0.1587 and P(Z < 2) = 0.9772. Subtracting these probabilities gives the probability that speed lies between 45 and 90 wpm which is 0.9772 - 0.1587 = 0.8185 or 81.85%.

Evaluate surprise for a typing speed exceeding 105 wpm

First calculate the z score for 105 wpm: Z = \((105 - 60) / 15 = 3\). P(Z < 3) = 0.9987 or 99.87%. So, around 0.13% of typists would type faster than 105 wpm. This is a very small percentage and hence it may be surprising.

Calculate probability for both randomly selected typists typing over 75 wpm

Z score for 75 wpm: Z = \((75 - 60) / 15 = 1\). P(Z < 1) = 0.8413 or 84.13%. The probability that typing speed is more than 75 wpm is the complement which is 1 - 0.8413 = 0.1587 or 15.87% . Since we’re selecting two typists independently, we multiply the probabilities: 0.1587 * 0.1587 = 0.0252 or 2.52%.

Determine the cut-off speed for the slowest 20%

To find z score for the lower 20% we look this up in the z-table and find it corresponds to approximately -0.84. To find the speed, we multiply the z value by the standard deviation and add to the mean: -0.84 * 15 + 60 = 47.4 wpm. So, anyone who types slower than this qualifies for the training.