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Chapter 8: Problem 2

FACTORY WoRKERS' WAGES According to the data released by the Chamber of Commerce of a certain city, the weekly wages of factory workers are normally distributed with a mean of \(\$ 600\) and a standard deviation of \(\$ 50\). What is the probability that a factory worker selected at random from the city makes a weekly wage a. Of less than \(\$ 600\) ? b. Of more than \(\$ 760 ?\) c. Between \(\$ 550\) and \(\$ 650\) ?

Short Answer

Expert verified
In conclusion: a. The probability of a factory worker having a weekly wage less than $600 is 0.5, or 50%. b. The probability of a factory worker having a weekly wage greater than $760 is 0.0007, or 0.07%. c. The probability of a factory worker having a weekly wage between \(550\) and \(650\) is 0.6826, or 68.26%.

Step by step solution

01

Find probability of wage less than $600

. Given the mean (\(\mu\)) is \(600 and the standard deviation (\)\sigma\() is \)50, calculate the z-score for a wage of $600. Here, the wage is equal to the mean wage, so the z-score is 0: \[z = \frac{600 - 600}{50} = 0\] Now, we look up the probability associated with a z-score of 0 in the standard normal table, which is 0.5. So, the probability of a factory worker having a weekly wage less than $600 is 0.5, or 50%.
02

Find probability of wage more than $760

. Calculate the z-score for a wage of $760: \[z = \frac{760 - 600}{50} = 3.2\] Now, we look up the probability associated with a z-score of 3.2 in the standard normal table, which is approximately 0.9993. However, we want the probability of a wage more than $760, so we will find the complement: \[1 - 0.9993 = 0.0007\] So, the probability of a factory worker having a weekly wage greater than $760 is 0.0007, or 0.07%.
03

Find probability of wage between \(550 and \)650

. Calculate the z-scores for wages of \(550 and \)650: \[z_1 = \frac{550 - 600}{50} = -1.0\] \[z_2 = \frac{650 - 600}{50} = 1.0\] Now, we look up the probabilities associated with z-scores of -1.0 and 1.0 in the standard normal table, which are approximately 0.1587 and 0.8413, respectively. To find the probability of a wage between \(550 and \)650, subtract the probabilities: \[0.8413 - 0.1587 = 0.6826\] So, the probability of a factory worker having a weekly wage between \(550 and \)650 is 0.6826, or 68.26%. In conclusion: a. The probability of a factory worker having a weekly wage less than $600 is 0.5, or 50%. b. The probability of a factory worker having a weekly wage greater than $760 is 0.0007, or 0.07%. c. The probability of a factory worker having a weekly wage between \(550 and \)650 is 0.6826, or 68.26%.

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Key Concepts

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

Normal Distribution
When investigating the weekly wages of factory workers, or any other type of data that follows a pattern where most observations cluster around a central peak, we're often dealing with what statisticians call a normal distribution.

This bell-shaped curve helps describe how the values of a variable are spread out or distributed. In the context of our factory workers' example, it means that most workers earn a wage around the mean, with fewer workers earning significantly more or less than that average.

Characteristics of a Normal Distribution

  • The mean, median, and mode of a normally distributed dataset are all equal.
  • It's symmetric about the mean鈥攈alf the values fall below the mean and half above it.
  • Approximately 68% of the data falls within one standard deviation of the mean, as seen with the wages between \(550 and \)650 in our exercise example.
  • Tails on either end of the curve approach the x-axis but never touch it, indicating that there are always a few individuals who earn much more or much less than average.
Understanding this distribution is fundamental to calculating probabilities for specific occurrences within a dataset.
Z-Score
Making sense of statistics involves more than just understanding the overall shape of data distributions; it involves comparing individual data points to the group. That's where the z-score comes in.

The z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point's score is identical to the mean score.

Calculating and Interpreting Z-Scores

  • To calculate the z-score of a particular value, you subtract the mean from the value and then divide that result by the standard deviation.
  • A positive z-score indicates a value above the mean, while a negative z-score signifies a value below the mean.
  • Higher absolute values of z-scores represent more unusual or extreme data points.
In the factory workers' wages example, we calculated both positive and negative z-scores to determine the rarity of certain wages.
Standard Deviation
The concept of standard deviation is at the heart of understanding variability within a set of data. In the context of our factory workers' wages, it helps to measure how dispersed the wages are from the average, or mean, wage.

The standard deviation provides a quantifiable indication of the spread of the scores. A small standard deviation means that the values are closely clustered around the mean; a large standard deviation suggests that the values are spread out over a wider range.

Importance in Probability

  • A lower standard deviation means more data points are close to the mean, as we might suppose with a tightly-controlled process in a factory setting.
  • When we compare different z-scores or calculate probabilities, the standard deviation is essential because it shows how much variation we can expect.
  • In our exercise, a standard deviation of $50 allowed us to calculate probabilities for wages at different thresholds.
The standard deviation is key in many areas of statistics, such as hypothesis testing, confidence interval construction, and of course, in everyday scenarios like determining wage probabilities.

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