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Chapter 6: Problem 62

Rockingham Corporation makes electric shavers. The life (period during which a shaver does not need a major repair) of Model J795 of an electric shaver manufactured by this corporation has a normal distribution with a mean of 70 months and a standard deviation of 8 months. The company is to determine the warranty period for this shaver. Any shaver that needs a major repair during this warranty period will be replaced free by the company. a. What should the warranty period be if the company does not want to replace more than \(1 \%\) of the shavers? b. What should the warranty period be if the company does not want to replace more than \(5 \%\) of the shavers?

Short Answer

Expert verified
The company should set the warranty period as roughly 51 months if it does not want to replace more than \(1\%\) of shavers, and around 57 months if it does not want to replace more than \(5\%\) of the shavers.

Step by step solution

01

Identify Parameters of Normal Distribution

The life of the electric shaver follows a normal distribution with a mean (碌) of 70 months and a standard deviation (蟽) of 8 months.
02

Calculate the Z-score for the Given Percentile

The Z-score is found from the standard normal distribution table, or using a calculator. \nFor \(1\%\) of shavers, the Z-score is approximately \( -2.33\). Likewise, for \(5\%\) of shavers, the Z-score is approximately \(-1.645\).
03

Convert Z-cores to Corresponding Warranty Periods

Use the formula X = 碌 + Z蟽 to convert the Z-score back to the original scale of measurement. This formula implicitly assumes the normal distribution of the measured variable.\nFor \(1\%\) of shavers, the warranty period (X) will be \(70 + (-2.33 * 8) = 51.36\) months.\nFor \(5\%\) of shavers, the warranty period (X) will be \(70 + (-1.645 * 8) = 56.84\) months.

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Key Concepts

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

Understanding Z-scores
A Z-score tells you how many standard deviations an element is from the mean of a distribution. It is a way to standardize scores on a common scale. For instance, when we talk about the life span of electric shavers, a Z-score can help us understand how unusual or typical a particular lifespan is.

The formula to calculate the Z-score is:\[Z = \frac{(X - \mu)}{\sigma}\]
  • \(X\) is the value of the data point (e.g., months until major repair)
  • \(\mu\) is the mean of the dataset
  • \(\sigma\) is the standard deviation of the dataset
The calculated Z-score can then be compared to a Z-table or used in software tools to find out the probability of a score occurring within a given roll of data or to identify scores in terms of standard deviations from the mean.
The Role of Mean and Standard Deviation
The mean and standard deviation are key components of the normal distribution. They allow us to summarize and understand data.

  • **The Mean (\(\mu\)):** This is the average of all data points. For the electric shavers, it is 70 months.
  • **The Standard Deviation (\(\sigma\)):** This measures how much variation or dispersion exists from the mean. For our shaver's lifespan, it's 8 months.
Together, the mean and standard deviation can tell us a lot about the "spread" and "center" of our data, which is extremely useful when making predictions like the warranty period. A smaller standard deviation means data points are closer to the mean, whereas a larger one indicates more spread out data.
Understanding Percentiles in the Context of Normal Distribution
Percentiles in a normal distribution help us determine at what point a certain percentage of the data falls below a certain value. In the context of electric shavers, knowing the 1% or 5% percentile helps determine a warranty period.

  • ***The 1% Percentile:*** This is the point where 1% of the data falls below, meaning the shaver lifespan surpasses this only in 1% of cases.
  • ***The 5% Percentile:*** Here, 5% of the shavers fail before reaching this lifespan.
Percentiles assist companies in deciding warranty periods by assessing these cutoff points; the calculation involves transforming these percentiles into Z-scores and converting them back into the original data scale.
Determining Warranty Period through Calculations
To set a warranty period, Rockingham Corporation wants to limit repairs so only a small percentage need attention within this time frame. Using Z-scores and percentiles, we convert back to the original time scale.

  • To ensure only 1% of shavers need replacement, look at the 1% Z-score of \(-2.33\).
  • To ensure only 5% need replacement, use a 5% Z-score of \(-1.645\).
The formula to find the warranty time \(X\) based on Z-score is:\[X = \mu + Z\sigma\]For 1%, calculate:\[ X = 70 + (-2.33 \times 8) = 51.36\] months.
For 5%, calculate:\[ X = 70 + (-1.645 \times 8) = 56.84 \] months.
By adjusting the warranty duration, the company can effectively limit how many products will need replacement under warranty.

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Most popular questions from this chapter

Alpha Corporation is considering two suppliers to secure the large amounts of steel rods that it uses. Company A produces rods with a mean diameter of \(8 \mathrm{~mm}\) and a standard deviation of \(.15 \mathrm{~mm}\) and sells 10,000 rods for \(\$ 400\). Company B produces rods with a mean diameter of \(8 \mathrm{~mm}\) and a standard deviation of \(.12 \mathrm{~mm}\) and sells 10,000 rods for \(\$ 460\). A rod is usable only if its diameter is between \(7.8 \mathrm{~mm}\) and \(8.2 \mathrm{~mm}\). Assume that the diameters of the rods produced by each company have a normal distribution. Which of the two companies should Alpha Corporation use as a supplier? Justify your answer with appropriate calculations.

Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees the refund of money or a replacement for any calculator that malfunctions within two years from the date of purchase. It is known from past data that despite all efforts, \(5 \%\) of the calculators manufactured by this company malfunction within a 2 -year period. The company recently mailed 500 such calculators to its customers. a. Find the probability that exactly 29 of the 500 calculators will be returned for refund or replacement within a 2-year period. b. What is the probability that 27 or more of the 500 calculators will be returned for refund or replacement within a 2 -year period? c. What is the probability that 15 to 22 of the 500 calculators will be returned for refund or replacement within a 2-year period?

Obtain the area under the standard normal curve a. to the right of \(z=1.43\) b. to the left of \(z=-1.65\) c. to the right of \(z=-.65\) d. to the left of \(z=.89\)

Major League Baseball rules require that the balls used in baseball games must have circumferences between 9 and \(9.25\) inches. Suppose the balls produced by the factory that supplies balls to Major League Baseball have circumferences normally distributed with a mean of \(9.125\) inches and a standard deviation of \(.06\) inch. What percentage of these baseballs fail to meet the circumference requirement?

Obtain the following probabilities for the standard normal distribution. a. \(P(z>-1.86)\) b. \(P(-.68 \leq z \leq 1.94)\) c. \(P(0 \leq z \leq 3.85)\) d. \(P(-4.34 \leq z \leq 0)\) e. \(P(z>4.82)\) f. \(P(z<-6.12)\)
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