The lifetimes of a certain type of car battery are normally distributed with a mean 5.9 years and standard deviation 0.4 year. The batteries are guaranteed to last at least 5 years. What proportion of batteries fail to meet the guarantee? In other words, what proportion of batteries last less than 5 years?

1.22% of batteries last less than 5 years

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

What proportion of batteries fail to meet the guarantee? In other words, what proportion of batteries last less than 5 years?

This is the pvalue of Z when X = 5. So

has a pvalue of 0.0122

1.22% of batteries last less than 5 years

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step-by-step explanation: