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Chapter 7: Problem 36

The accounting department at Weston Materials Inc., a national manufacturer of unattached garages, reports that it takes two construction workers a mean of 32 hours and a standard deviation of 2 hours to erect the Red Barn model. Assume the assembly times follow the normal distribution. a. Determine the \(z\) values for 29 and 34 hours. What percent of the garages take between 32 hours and 34 hours to erect? b. What percent of the garages take between 29 hours and 34 hours to erect? c. What percent of the garages take 28.7 hours or less to erect? d. Of the garages, \(5 \%\) take how many hours or more to erect?

Short Answer

Expert verified
a) 34%, b) 86.64%, c) 4.95%, d) 35.29 hours.

Step by step solution

01

Calculate the z-score for 29 hours

To calculate the z-score for 29 hours, use the formula: \[ z = \frac{X - \mu}{\sigma} \] where \( X = 29 \), \( \mu = 32 \), and \( \sigma = 2 \). Thus the z-score is: \( z = \frac{29 - 32}{2} = -1.5 \).
02

Calculate the z-score for 34 hours

Using the same z-score formula as in Step 1, substitute \( X = 34 \): \[ z = \frac{34 - 32}{2} = 1 \].
03

Find the percentage between 32 and 34 hours

The z-score for 32 hours is 0 (since it is the mean). Use the z-scores from Steps 1 and 2: \( z = 0 \) to \( z = 1 \). Using a z-table, the probability from \( z = 0 \) to \( z = 1 \) is approximately 34\/100 = 0.34, or 34%.
04

Find the percentage between 29 and 34 hours

Use the z-scores \( z = -1.5 \) and \( z = 1 \). From a z-table, the probability from \( z = -1.5 \) to \( z = 1 \) is approximately 0.9332 - 0.0668 = 0.8664, or 86.64%.
05

Calculate the percentage for 28.7 hours or less

Calculate the z-score for 28.7 hours: \( z = \frac{28.7 - 32}{2} = -1.65 \). From a z-table, the probability below \( z = -1.65 \) is approximately 0.0495, or 4.95%.
06

Determine hours for the top 5%

Find the z-score for the top 5\( \% \) (right-side tail), which is approximately \( z = 1.645 \) from a z-table. Use the formula \( X = \mu + z\sigma \): \( X = 32 + 1.645(2) = 35.29 \). Therefore, 5\( \% \) take 35.29 hours or more to erect.

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Key Concepts

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

Z-score
The z-score is a numerical measurement that describes a value's relationship to the mean of a set of values. In simpler terms, it tells us how far away, in terms of standard deviations, a particular data point or observation is from the mean.
To calculate the z-score, use the formula: \[ z = \frac{X - \mu}{\sigma} \]where:
  • \( X \) is the data point in question.
  • \( \mu \) is the mean of the dataset.
  • \( \sigma \) is the standard deviation of the dataset.
For instance, to compute the z-score for 29 hours when the mean is 32 and the standard deviation is 2, it becomes:\[ z = \frac{29 - 32}{2} = -1.5 \].
This tells us that 29 hours is 1.5 standard deviations below the mean.
Probability
Probability in a statistical context refers to the measure of the likelihood that a given event will occur. The probability can range from 0 (impossible event) to 1 (certain event). In the context of the normal distribution, this probability is often represented as the area under the curve.
When given a z-score, you can use a z-table to look up the probability associated with it. For example, a z-score from 0 to 1 corresponds to a probability of 34%, meaning that 34% of all data points are between these two scores.
This is crucial for interpreting ranges in data, such as determining the likelihood that a garage construction will take between 32 and 34 hours.
Standard Deviation
Standard deviation is a measurement of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
In our exercise, the standard deviation of 2 hours means that most construction times deviate from the average time of 32 hours by about 2 hours. This gives us a clear indication of the reliability and consistency of the construction time.
It is an essential element in the calculation of the z-score and determining probabilities.
Mean
The mean, often referred to as the average, is the sum of all the values in a dataset divided by the number of values. In statistical normal distributions, the mean is located at the center of the bell curve.
In our case, the mean time it takes to erect the garages is 32 hours. This value serves as the reference point for calculating z-scores and probabilities. The mean is crucial because it serves as a measure of the dataset's central tendency, offering insight into where most values cluster around.
Understanding the mean helps in understanding what is typical or average within a given dataset.

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Most popular questions from this chapter

A normal population has a mean of 20.0 and a standard deviation of \(4.0 .\) a. Compute the \(z\) value associated with 25.0 . b. What proportion of the population is between 20.0 and \(25.0 ?\) c. What proportion of the population is less than \(18.0 ?\)

Most four-year automobile leases allow up to 60,000 miles. If the lessee goes beyond this amount, a penalty of 20 cents per mile is added to the lease cost. Suppose the distribution of miles driven on four-year leases follows the normal distribution. The mean is 52,000 miles and the standard deviation is 5,000 miles. a. What percent of the leases will yield a penalty because of excess mileage? b. If the automobile company wanted to change the terms of the lease so that \(25 \%\) of the leases went over the limit, where should the new upper limit be set? c. One definition of a low-mileage car is one that is 4 years old and has been driven less than 45,000 miles. What percent of the cars returned are considered low-mileage?

Fast Service Truck Lines uses the Ford Super Duty F-750 exclusively. Management made a study of the maintenance costs and determined the number of miles traveled during the year followed the normal distribution. The mean of the distribution was 60,000 miles and the standard deviation was 2,000 miles. a. What percent of the Ford Super Duty F-750s logged 65,200 miles or more? b. What percent of the trucks logged more than 57,060 but less than 58,280 miles? c. What percent of the Fords traveled 62,000 miles or less during the year? d. Is it reasonable to conclude that any of the trucks were driven more than 70,000 miles? Explain.

Many retail stores offer their own credit cards. At the time of the credit application, the customer is given a \(10 \%\) discount on the purchase. The time required for the credit application process follows a uniform distribution with the times ranging from 4 minutes to 10 minutes. a. What is the mean time for the application process? b. What is the standard deviation of the process time? c. What is the likelihood a particular application will take less than 6 minutes? d. What is the likelihood an application will take more than 5 minutes?

A normal distribution has a mean of 50 and a standard deviation of 4. a. Compute the probability of a value between 44.0 and 55.0 . b. Compute the probability of a value greater than 55.0 . c. Compute the probability of a value between 52.0 and 55.0 .
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