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Chapter 7: Problem 43

Management at Gordon Electronics is considering adopting a bonus system to increase production. One suggestion is to pay a bonus on the highest \(5 \%\) of production, based on past experience. Past records indicate weekly production follows the normal distribution. The mean of this distribution is 4,000 units per week and the standard deviation is 60 units per week. If the bonus is paid on the upper \(5 \%\) of production, the bonus will be paid on how many units or more?

Short Answer

Expert verified
The bonus will be paid on production of 4099 units or more.

Step by step solution

01

Understand the Problem

We need to find the production level above which the top 5% of weekly production falls. Since the production follows a normal distribution with a mean of 4000 units and a standard deviation of 60 units, we can use these parameters to find the cutoff point for the highest 5%.
02

Identify the Z-score for the Top 5%

The top 5% in a normal distribution corresponds to the 95th percentile (since it includes 5% in the tail). We need to find the Z-score that marks the point where 95% of the data lies below it. Using standard Z-score tables, we find the Z-score for the 95th percentile is approximately 1.645.
03

Calculate the Production Level for the Bonus

We use the Z-score formula for a normal distribution: \[ X = ext{mean} + Z imes ext{standard deviation} \]Substitute the known values: \[ X = 4000 + 1.645 imes 60 \]Calculate this to find the production level cutoff.
04

Perform the Calculation

Substitute and calculate:\[ X = 4000 + 1.645 imes 60 = 4000 + 98.7 = 4098.7 \]Rounding to the nearest full unit, the production level is 4099 units.

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Key Concepts

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

Percentile Rank
When discussing normal distributions, percentile rank is an important concept. Percentile rank indicates the relative standing of a value within a set of data. If a score is at the 95th percentile, it means that it is higher than 95% of the other observations. In the context of our exercise, we are interested in the top 5% of production levels. This corresponds to the 95th percentile since the highest 5% surpasses 95% of the other data.
To determine the cutoff for the bonus system, we have to find the production value that marks this top segment.
The percentile rank helps in setting thresholds such as bonuses or grades in practical scenarios, making it a widely used statistical tool.
Standard Deviation
Standard deviation is a measure that tells us how data is dispersed around the mean in a normal distribution. In simple terms, it provides insights into the spread of the data points.
In the given exercise, the standard deviation is used to understand how much individual production weeks can vary from the average of 4,000 units.
  • A small standard deviation indicates that the data points tend to be very close to the mean, suggesting consistent production levels.
  • A larger standard deviation would imply more variability in production each week.
By accounting for standard deviation, we can effectively calculate the production level that marks the threshold for paying bonuses.
Z-Score Calculation
The z-score is a statistical measurement that helps in understanding where a particular value lies within a normal distribution. It indicates how many standard deviations a value is from the mean.
In our exercise, we use z-score calculation to determine the production level threshold for the top 5% of production outputs.
To find this threshold, we look up the z-score corresponding to the 95th percentile in z-score tables, which is approximately 1.645.
The formula used:\[ X = \text{mean} + Z \times \text{standard deviation} \]where:
  • \(X\) is the production level we are solving for
  • The mean is given as 4,000 units
  • The standard deviation is 60 units
  • The z-score is 1.645
By plugging these values into the formula, we determine that the production level required to qualify for the bonus is 4,099 units after rounding. This calculation helps businesses set precise and efficient decision thresholds.

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Most popular questions from this chapter

The mean of a normal probability distribution is 400 pounds. The standard deviation is 10 pounds. a. What is the area between 415 pounds and the mean of 400 pounds? b. What is the area between the mean and 395 pounds? c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

In economic theory, a "hurdle rate" is the minimum return that a person requires before he or she will make an investment. A research report says that annual returns from a specific class of common equities are distributed according to a normal distribution with a mean of \(12 \%\) and a standard deviation of \(18 \%\). A stock screener would like to identify a hurdle rate such that only 1 in 20 equities is above that value Where should the hurdle rate be set?

Suppose the Internal Revenue Service reported that the mean tax refund for the year 2017 was \(\$ 2,800 .\) Assume the standard deviation is \(\$ 450\) and that the amounts refunded follow a normal probability distribution. a. What percent of the refunds are more than \(\$ 3,100 ?\) b. What percent of the refunds are more than \(\$ 3,100\) but less than \(\$ 3,500 ?\) c. What percent of the refunds are more than \(\$ 2,250\) but less than \(\$ 3,500 ?\)

Recently the United States Department of Agriculture issued a report (http://www.cnpp. usda.gov/sites/default/files/CostoffoodMar2015.pdf) indicating a family of four spent an average of about \(\$ 890\) per month on food. Assume the distribution of food expenditures for a family of four follows the normal distribution, with a standard deviation of \(\$ 90\) per month. a. What percent of the families spend more than \(\$ 430\) but less than \(\$ 890\) per month on food? b. What percent of the families spend less than \(\$ 830\) per month on food? c. What percent spend between \(\$ 830\) and \(\$ 1,000\) per month on food? d. What percent spend between \(\$ 900\) and \(\$ 1,000\) per month on food?

Assume that the hourly cost to operate a commercial airplane follows the normal distribution with a mean of \(\$ 2,100\) per hour and a standard deviation of \(\$ 250 .\) What is the operating cost for the lowest \(3 \%\) of the airplanes?
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