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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 \(60 \mathrm{wpm}\) and standard deviation 15 wpm ("The Effects of Split Keyboard Geometry on Upper body Postures," Ergonomics [2009]: 104-1I1). 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 typing 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?

Step by step solution

01

Calculate the Probability for Typist's Speed is at most 60 wpm

Since 60 wpm is equal to the mean, a randomly selected typist's speed being at most 60 wpm is equal to P(Z ≤ 0) = 0.5 or 50%. P(Z < 0) is also equal to 0.5 or 50%.

02

Calculate the Probability for Typist's Speed is between 45 and 90 wpm

First standardize these scores using the z-score formula, \( z = (x - \mu)/\sigma\). For 45, \( z = (45 - 60)/15 = -1 \) and for 90, \( z = (90 - 60)/15 = 2 \). Now, find the probabilities associated with these z-scores from a standard normal distribution table, \(P(-1 \leq Z \leq 2)\), or we need to find \(P(Z \leq 2) - P(Z \leq -1) = 0.9772 - 0.1587 = 0.8185 \) or 81.85%.

03

Discuss about Typist with Speed exceeding 105 wpm

Standardize 105 to z-score, \( z = (105 - 60)/15 = 3 \). From the standard normal distribution table, \( P(Z \leq 3) = 0.9987 \) or \~99.87%. \(P(Z > 3) = 1 - 0.9987 = 0.0013 \) or 0.13%. This means the probability that a typist from this population typing faster than 105 wpm is extremely small, so one would indeed be surprised to find such a typist.

04

Probability that both Typists typing speed exceeds 75 wpm

The speed of 75 wpm corresponds to \( z = (75 - 60)/15 = 1 \). The probability that a single typist types faster than 75 wpm is \( P(Z > 1) = 1 - P(Z \leq 1) = 1 - 0.8413 = 0.1587 \) or 15.87%. Since two typists are selected independently, the needed probability is \(0.1587 * 0.1587 = 0.0252 \) or \~2.52%.

05

Typing speeds qualifying individuals for the training

We need to find the typing speed that marks the cutoff for slowest 20% typists. We need to find the z-score associated with the 20th percentile in the standard normal distribution. From the tables, the z-score for 0.20 is approximately -0.8416. We then convert this to a typing speed using the formula: \( X = Z * \sigma + \mu = -0.8416 * 15 + 60 = 47.4 \approx 47 \) wpm. Therefore, typing speeds of under 47 wpm would qualify for the training.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

Normal distribution is a fundamental concept in statistics that describes how data points are spread out. It is often called a "bell curve" because of its symmetrical, bell-shaped appearance when graphed.
This distribution is characterized by two parameters:

• The mean (\(\mu\)) which determines the center of the curve.
• The standard deviation (\(\sigma\)), which controls the width of the curve.

In a normal distribution, most of the data points tend to cluster around the mean, with fewer and fewer data points appearing as you move away from the mean in both directions. This property allows us to make statements about probabilities and how likely it is to find values within a certain range. For example, around 68% of data falls within one standard deviation of the mean in a normal distribution.

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean.
The formula to calculate a z-score is:

• \(z = \frac{x - \mu}{\sigma}\)

where:

• \(x\) is the value you want to convert.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation of the distribution.

A z-score indicates how many standard deviations an element is from the mean. A z-score of 0 indicates that the data point's score is identical to the mean. Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean. Z-scores are crucial in identifying outliers and understanding probabilities in a normal distribution.

Probability

Probability refers to the measure of the likelihood that an event will occur. It is always expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 indicates certainty.
In the context of statistics and normal distribution, we often want to find the probability of a particular outcome or range of outcomes occurring. This is done using the standard normal distribution table (z-table), which helps us to find the probability that a statistic is less than or equal to a specific z-score.
The rules of probability can be intuitive:

• The probability of any single event is the sum of the probabilities of all possible outcomes.
• Non-overlapping events' probabilities will add up.

For instance, if you know the probability that a typist's speed is below a certain wpm, then the probability of it being above is simply the complement that adds up to 1.

Percentiles

Percentiles are a way of expressing statistics that tell us about the relative standing of a particular value within a dataset. If a score is in the 75th percentile, it means it is higher than 75% of the other scores in the dataset.
To find what percentile a specific value corresponds to, we use the z-score and refer to the standard normal distribution table to find the area to the left of that z-score. This area directly corresponds to the percentile.
The key points about percentiles:

• They range from 0 to 100, representing the full distribution of data.
• Percentiles help to understand how a particular value compares to the rest of the data.

When it comes to making decisions, like identifying the slowest typists for extra training, percentiles are very useful. By knowing the cutoff for the 20th percentile, you know which typists fall in the bottom 20% of speeds.