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Chapter 8: Problem 37

The deluxe model hair dryer produced by Roland Electric has a mean expected lifetime of 24 mo with a standard deviation of 3 mo. Find a bound on the probability that one of these hair dryers will last between 20 and 28 mo.

Short Answer

Expert verified
The probability that a deluxe model hair dryer produced by Roland Electric will last between 20 and 28 months is approximately 81.64%. This is calculated using the properties of a normal distribution, with a mean expected lifetime of 24 months, a standard deviation of 3 months, and respective Z-scores of -1.33 for 20 months and 1.33 for 28 months.

Step by step solution

01

Identify given information

The given information is: Mean expected lifetime (碌) = 24 months Standard deviation (蟽) = 3 months Lower bound (A) = 20 months Upper bound (B) = 28 months
02

Calculate Z-scores for A and B

To calculate the Z-scores for A and B, we will use the Z-score formula: Z = \(\frac{X - 碌}{蟽}\) For A (20 months): \(Z_A = \frac{20 - 24}{3} = \frac{-4}{3} = -1.33\) For B (28 months): \(Z_B = \frac{28 - 24}{3} = \frac{4}{3} = 1.33\) Now, we have Z-scores Z_A = -1.33 and Z_B = 1.33.
03

Find the probability using Z-scores and standard normal distribution table

To find the probability, we need to consult the standard normal distribution table for the corresponding probabilities of Z_A and Z_B. This table typically provides the probability from the left tail (Z-score of -鈭�) up to a given Z-score. For Z_A = -1.33, we find that the probability is approximately 0.0918. For Z_B = 1.33, we find that the probability is approximately 0.9082. Now, to find the probability between Z_A and Z_B, we will subtract the probability of Z_A from Z_B: Probability (P) = P(Z_B) - P(Z_A) P = 0.9082 - 0.0918 P = 0.8164 Therefore, there is an approximately 81.64% chance that the lifetime of a deluxe model hair dryer will last between 20 and 28 months.

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