a manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.3 years, and…

a manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.3 years, and standard deviation of 1.6 years.\n\nthe 1% of items with the shortest lifespan will last less than how many years?\n\nround your answer to one decimal place.

a manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.3 years, and standard deviation of 1.6 years.\n\nthe 1% of items with the shortest lifespan will last less than how many years?\n\nround your answer to one decimal place.

Answer

Explanation:

Step1: Identify given parameters

The mean is $\mu = 5.3$ and the standard deviation is $\sigma = 1.6$.

Step2: Determine the target percentile

The shortest $1%$ corresponds to the $1^{st}$ percentile, so $P(X < x) = 0.01$.

Step3: Find the corresponding z-score

Using a standard normal distribution table, the z-score for $0.01$ is approximately $-2.326$. $$z \approx -2.326$$

Step4: Calculate the lifespan value

Use the formula $x = \mu + z\sigma$ to find the specific lifespan. $$x = 5.3 + (-2.326)(1.6)$$

Step5: Perform the arithmetic

Calculate the product and add it to the mean. $$x = 5.3 - 3.7216 = 1.5784$$

Step6: Round to one decimal place

Round the final result as requested. $$x \approx 1.6$$

Answer:

1.6