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Chapter 9: Problem 570

Let \(\mathrm{X}\) be a normally distributed random variable representing the hourly wage in a certain craft. The mean of the hourly wage is \(\$ 4.25\) and the standard deviation is \(\$ .75 .\) (a) What percentage of workers receive hourly wages between \(\$ 3.50\) and \(\$ 4.90 ?\) (b) What hourly wage represents the 95 th percentile?

Short Answer

Expert verified
(a) The percentage of workers receiving an hourly wage between $3.50 and $4.90 is approximately \(68.79\% \). (b) The hourly wage representing the 95th percentile is approximately $\(5.64 \).

Step by step solution

01

Calculate the z-scores for the given wages

In order to find the percentage of workers with wages between \(3.50 and \)4.90, we have to find the z-scores for both these wages. A z-score is calculated using the formula: z = \(\frac{X - 渭}{蟽}\), where X is the data point, 渭 is the mean, and 蟽 is the standard deviation. For X = $3.50: z1 = \(\frac{3.50 - 4.25}{0.75}\) For X = $4.90: z2 = \(\frac{4.90 - 4.25}{0.75}\) Step 2: Finding the percentages using z-scores
02

Use a z-table to find the percentages for z1 and z2

Once we have the z-scores, we can use a standard normal distribution table (z-table) to find the percentages corresponding to each z-score. Look up the values of z1 and z2 in the z-table. Step 3: Calculating the percentage of workers
03

Subtract the percentages corresponding to z1 and z2

To find the percentage of workers receiving hourly wages between \(3.50 and \)4.90, subtract the percentage corresponding to z1 from the percentage corresponding to z2. Percentage of workers = Percentage (z2) - Percentage (z1) (a) Report the percentage of workers receiving hourly wages between \(3.50 and \)4.90. For part (b), we have to find the hourly wage representing the 95th percentile. Step 4: Find z-score corresponding to 95th percentile
04

Use a z-table to find the z-score corresponding to the 95th percentile

Look up the 95th percentile in a z-table. It should give you a z-score close to the required percentile. Step 5: Calculate the hourly wage using the z-score
05

Use the z-score formula to find the hourly wage

Once we have the z-score for the 95th percentile, we can use the z-score formula to find the corresponding hourly wage. Rearranging the formula, we have: X = 渭 + z * 蟽 Plug in the values of z, 渭, and 蟽, and calculate the hourly wage. (b) Report the hourly wage representing the 95th percentile.

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Key Concepts

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

Understanding Z-Scores
Z-scores are a simple yet powerful way to understand how far a particular data point is from the mean in a distribution. A z-score tells us how many standard deviations a point is from the mean. This is important for comparing different data points and understanding where they stand in the distribution.

To calculate a z-score, use the formula:
  • \( z = \frac{X - \mu}{\sigma} \)
  • \( X \) is the data point (e.g., an hourly wage)
  • \( \mu \) is the mean of the distribution
  • \( \sigma \) is the standard deviation
By converting values to z-scores, we can use standard z-tables to find probabilities and percentiles within the distribution.
The Role of Percentiles
Percentiles are values that divide a dataset into 100 equal parts. They help you understand the relative standing of a value within a dataset. For instance, if you are at the 95th percentile in height, you are taller than 95% of the population.

In the context of normal distributions:
  • Percentiles help determine thresholds for data points.
  • They are useful for finding outliers or points of interest.
If tasked with finding the 95th percentile wage, first find the z-score corresponding to 95% using a z-table. Then, use this z-score to calculate the value within the distribution.
Explaining Standard Deviation
Standard deviation is a measure of the spread or dispersion of a set of data. In a normal distribution, it indicates how much the data points deviate from the mean. A small standard deviation means data points are close to the mean, while a large standard deviation indicates the opposite.

Understanding standard deviation helps in several ways:
  • It defines the shape of the distribution curve.
  • Helps calculate z-scores.
  • Provides insight into variability.
In practical scenarios like wage distribution, knowing the standard deviation aids in understanding wage variability among workers.
What is the Mean?
The mean, also known as the average, is the sum of all data points divided by the number of data points. It provides a central or typical value within a dataset, reflecting the overall level of the data.

The mean is crucial for:
  • Comparing individuals or groups within a dataset.
  • Functioning as a baseline for calculations like the z-score.
  • Offering insights into the distribution's center.
In wage studies, the mean wage is the typical earning expected in that field, serving as a reference point for understanding other wages in the distribution.

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Most popular questions from this chapter

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