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Chapter 6: Problem 24

According to Current Population Reports, self-employed individuals in the United States work an average of 44.6 hours per week, with a standard deviation of \(14.5 .\) If this variable is approximately normally distributed, find the proportion of the self-employed who work more than 40 hours per week. Sketch a graph, and mark off on it \(44.6,40,\) and the region to which the answer refers.

Short Answer

Expert verified
Approximately 62.46% of self-employed individuals work more than 40 hours per week.

Step by step solution

01

Understand the Normal Distribution

We're given a normal distribution of self-employed weekly work hours with a mean (9) of 44.6 hours and a standard deviation (1) of 14.5 hours. We need to find the proportion of workers who work more than 40 hours.
02

Convert to the Standard Normal Distribution

We need to convert 40 hours to a z-score using the formula: \[z = \frac{X - \mu}{\sigma}\]where \(X\) is 40, \(\mu\) is 44.6, and \(\sigma\) is 14.5. Substituting the values, we have:\[z = \frac{40 - 44.6}{14.5} = \frac{-4.6}{14.5} \approx -0.317\]
03

Use the Z-table to Find Proportion

Using the z-score obtained (-0.317), we consult the z-table to find the probability that a normal random variable is less than 40 hours. The z-table gives the probability for z < -0.317, approximately 0.3754.
04

Find the Proportion Greater than 40 Hours

To find the proportion of individuals working more than 40 hours, subtract the cumulative probability from 1:\[P(X > 40) = 1 - P(Z < -0.317) = 1 - 0.3754 = 0.6246\]This means that approximately 62.46% of self-employed individuals work more than 40 hours per week.
05

Sketch the Graph

Draw a normal distribution curve with a mean of 44.6. Mark the 40-hour point on the curve below the mean. Shade the area to the right of the 40-hour mark, as this area represents the proportion of individuals working more than 40 hours, corresponding to 62.46%.

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Key Concepts

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

Z-Score Calculation
The concept of the z-score is essential in understanding how individual data points relate to the mean of a data set. Essentially, it tells us how many standard deviations a particular value (in this case, work hours) is from the mean.The formula for calculating the z-score is:\[ z = \frac{X - \mu}{\sigma} \]Where:
  • \( X \) is the data point of interest, here, the 40 hours worked.
  • \( \mu \) is the mean of the distribution, which is 44.6 hours.
  • \( \sigma \) is the standard deviation, given as 14.5 hours.
Substituting these values into the formula gives us:\[ z = \frac{40 - 44.6}{14.5} = \frac{-4.6}{14.5} \approx -0.317 \]A z-score of -0.317 means that 40 hours is 0.317 standard deviations below the mean. This calculation helps us determine where 40 hours falls in relation to the average work week for self-employed individuals.
Cumulative Probability
Cumulative probability asks us to find the probability that a random variable is less than or equal to a certain value. When we use the z-table, it provides us with this cumulative probability for values less than a specific z-score. With our problem, the z-score calculated was -0.317. To find the probability of a self-employed individual working less than 40 hours, consult the z-table. For a z-score of -0.317, the corresponding cumulative probability is approximately 0.3754. This probability reflects the portion of the bell curve to the left of 40 hours. When considering the context of the problem, this 0.3754 means about 37.54% of self-employed individuals work less than 40 hours per week. This information is helpful because the z-table provides a way to understand how common or uncommon a data point is based on the normal distribution.
Self-Employed Work Hours
When examining the work hours of self-employed individuals, we rely on data like averages and standard deviations to understand typical work patterns. For our data set:
  • The mean is 44.6 hours, which signifies the average work week.
  • The standard deviation, 14.5 hours, shows the variability in work hours.
In this scenario, we found that a significant percentage, about 62.46%, of self-employed individuals exceed the 40-hour work week. These results highlight a tendency among self-employed workers towards longer work weeks, compared to perhaps traditional employment which often centers around a 40-hour framework. Understanding these statistics can help provide insights into economic behaviors and aid in planning for self-employed individuals and policymakers.

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Most popular questions from this chapter

SAT math scores follow a normal distribution with an approximate \(\mu=500\) and \(\sigma=100\). Also ACT math scores follow a normal distrubution with an approximate \(\mu=21\) and \(\sigma=4.7\). You are an admissions officer at a university and have room to admit one more student for the upcoming year. Joe scored 600 on the SAT math exam, and Kate scored 25 on the ACT math exam. If you were going to base your decision solely on their performances on the exams, which student should you admit? Explain.

The Internet site www.ItsJustLunch .com advertises itself as a dating service for busy professionals that has set up over two million first dates for lunch or drinks after work. An advertisement for this site stated that a survey of their users found that a woman has chance 1 in 8 of a second date if she has not heard from the man within 24 hours of their first date. On Saturday, Shawna had a luncheon date with Jack and a dinner date with Lawrence. By Sunday evening she had not heard from either of them. Based on the information claimed by www.ItsJustLunch.com, construct a table with the probability distribution of \(X=\) the number of these men 2) with whom she has a second date. (Source: \((0,1,\) or Background information from www.ItsJustLunch.com.)

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The normal distribution for women's height in North America has \(\mu=65\) inches, \(\sigma=3.5\) inches. Most major airlines have height requirements for flight attendants (www.cabincrewjobs.com). Although exceptions are made, the minimum height requirement is 62 inches. What proportion of adult females in North America are not tall enough to be a flight attendant?
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