Question

Suppose that the distribution of net typing rate in words per minute (wpm) for experienced typists can be approximated by a normal curve with mean \(60 \mathrm{wpm}\) and standard deviation 15 wpm ("Effects of Age and Skill in Typing", Journal of Experimental Psychology [1984]: \(345-371\) ). a. What is the probability that a randomly selected typist's net rate is at most 60 wpm? less than 60 wpm? b. What is the probability that a randomly selected typist's net rate is between 45 and 90 wpm? c. Would you be surprised to find a typist in this population whose net rate exceeded 105 wpm? (Note: The largest net rate in a sample described in the paper is 104 wpm.) d. Suppose that two typists are independently selected. What is the probability that both their typing rates 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?

Short answer

a. The probability is 0.5 for both at most 60 wpm and less than 60 wpm. \n\nb. The probability is approximately 0.8185. \n\nc. Yes, it would be extremely surprising. \n\nd. The probability is approximately 0.0252. \n\ne. Typing speeds up to about 47.4 wpm would qualify for the training.

Step-by-step solution

Convert values to z-scores

To convert a value to a z-score, use the formula \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.

Use standard normal table to find probability

After converting to a z-score, the value of the standard normal table at the z-score gives the probability that a randomly selected value from the distribution does not exceed \(x\) (is less than or equal to \(x\)). If the question asks for the probability that the value is strictly less than \(x\), subtract a small amount from \(x\) before converting to a z-score.

Answer each part of the question

a. For 60 wpm, the z-score is 0 (because 60 is the mean, so it is 0 standard deviations away from the mean). The value of the standard normal table at 0 is 0.5, so the probability that a randomly selected typist's net rate is at most 60 wpm is 0.5. The same answer applies for less than 60 wpm, as it is practically impossible for there to be a value exactly equal to 60 wpm. \n\nb. For 45 wpm, the z-score is \(\frac{45-60}{15} = -1\), and for 90 wpm, the z-score is \(\frac{90-60}{15} = 2\). The value of the standard normal table at -1 is about 0.1587 and at 2 is about 0.9772. Thus, the probability that a randomly selected typist's net rate is between 45 and 90 wpm is the difference of these two probabilities, which is \(0.9772 - 0.1587 = 0.8185\). \n\nc. For 105 wpm, the z-score is \(\frac{105-60}{15} = 3\). The value of the standard normal table at 3 is about 0.9987, which implies that less than 0.1% of typists have a net rate more than 105 wpm. Hence, it would be extremely surprising to find a typist in this population whose net rate exceeded 105 wpm. \n\nd. The probability that one typist has a rate that exceeds 75 wpm is \(1 - \) the value of the standard normal table at \(\frac{75-60}{15} = 1\), which is \(1 - 0.8413 = 0.1587\). Because the two typists are independently selected, the probability that both their typing rates exceed 75 wpm is this value squared, \(0.1587^2 = 0.0252\). \n\ne. The slowest 20% of the typists are those whose net rates do not exceed the value of \(x\) at which the value of the standard normal table is 0.20. This value is about -0.84. Therefore, the corresponding typing speed is -0.84 standard deviations away from the mean, or \(60 - 0.84*15 = 47.4\) wpm.

Check your answers

It is good practice to check your answers. For example, for parts b and c, you could check if the given wpm rates are within the likely range of the distribution.

Key concepts

Z-Score

Understanding the z-score is crucial for students dealing with normal distribution problems. The z-score, also known as the standard score, is a numerical measurement that describes a value's relationship to the mean of a group of values. It's calculated using the formula:
\[ z = \frac{x - \mu}{\sigma} \]
where \( x \) is the value in question, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation. If the z-score is 0, it means the value is exactly at the mean. If it's positive, the value is above the mean, and if it's negative, the value is below the mean.
For example, if typists have an average typing rate of 60 words per minute, and one typist types at 75 words per minute, their z-score will tell us how many standard deviations above the mean they are. In our case study, a typing rate of 60 wpm has a z-score of 0 since it's exactly at the mean. By converting raw scores to z-scores, we standardize them and can more easily compare different data points or find probabilities using the standard normal distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution that's used when working with z-scores. It has a mean of 0 and a standard deviation of 1. What's useful about it is that it allows us to compare scores from different normal distributions by converting them to a common scale.
When values are transformed into z-scores, we can use a standard normal table (also called a z-table) to find probabilities. This table shows the probability that a normally distributed random variable will be less than or equal to a given z-score. In practical terms, this means that if you have a z-score, you can look it up in the table to find the probability of obtaining a value within that standard deviation range.
In our example, if the probability of a typist typing less than 60 wpm is sought, we use the z-score of 0 and find its corresponding probability, which is 0.5. This means there's a 50% chance that a random typist will type fewer than 60 words per minute, which aligns with the fact that 60 wpm is the mean—half the typists will be slower, and half will be faster, at least theoretically.

Probability Distribution

A probability distribution assigns a probability to each possible outcome of a random variable. In the context of normal distribution, it tells us how the typing speeds of typists are dispersed around the mean, which is 60 wpm in our scenario. The shape of the distribution is the familiar bell curve: This means that values closer to the mean (60 wpm) are more likely to occur, while those further from the mean are less likely.
The probability of finding a randomly selected typist with a specific typing rate can be calculated using the areas under the curve. For instance, to find the probability that a typist types between 45 and 90 wpm, we would look at the area under the curve between these two values. The wider the range of values, the more area under the curve, and therefore, the higher the probability.
Probability distributions are integral to statistics because they provide a mathematical framework for quantifying the uncertainty associated with random variables. In educational settings, grasping this concept helps students to assess the likelihood of different outcomes and make informed predictions based on the given data.