The average life of the RGC type of motor is 8 years, with a...

The average life of the RGC type of motor is 8 years, with a standard deviation of 1.5 years. If the manufacturer is willing to replace only $5 \%$ of the motors that fails, how long is the guarantee that he can offer? Assume that the lives of the motors follow a normal distribution.

Key Concepts

Z-scores

Z-scores are standardized values that signify the number of standard deviations a data point is from the mean of the distribution. They are used to convert individual data points into a common scale, which makes it easier to calculate probabilities and percentiles using the standard normal distribution.

Percentiles

Percentiles are values below which a certain percentage of observations fall. For instance, the 5th percentile indicates that 5% of the data lies below that value. This concept is essential in many statistical applications, such as setting warranties or guarantees, where a threshold is chosen to cover only a specified portion of the data.

Reliability Analysis

Reliability analysis involves the use of statistical methods and distributions, like the normal distribution, to assess the lifespan and performance of products. It helps in determining the likelihood of failure within a particular time frame and is crucial for making informed decisions about warranties and maintenance policies.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by its mean and standard deviation and is widely used to model natural phenomena. In many applications, it helps in understanding the probability of a variable falling within certain ranges about the mean.

Standard Deviation

Standard deviation measures the dispersion of data points around the mean. In the context of the normal distribution, it determines the spread of data and is crucial for quantifying how much variation exists from the average value. It directly influences the probability estimates of events occurring within specific intervals.

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