Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 4.9 years and standard deviation of 1.5 years. If 7 items are picked at random, 3% of the time their lifespan will be above what value?

Answer 1

To calculate the lifespan value above which 3% of the items fall, we identify the z-score for the top 3%, 1.88, and use the formula X = mean + (z × standard deviation) which yields 7.72 years.

Step-by-step explanation:

To find the value above which the lifespan of the items will be 3% of the time, we need to determine the z-score that corresponds to the upper 3% tail of the standard normal distribution. Once we have the z-score, we can then use the mean and standard deviation of the item's lifespan to find the particular lifespan value.

Firstly, using a standard normal distribution table or a calculator with normal distribution functions, we find that the z-score corresponding to the upper 97% is typically around 1.88. This is because there is 3% in the upper tail.

The formula to convert the z-score to the specific value (X) in the context of the item's lifespan is:

X = mean + (z × standard deviation)

For the given mean (μ) of 4.9 years and standard deviation (σ) of 1.5 years, we calculate:

X = 4.9 + (1.88 × 1.5)

By performing the calculation:

X = 4.9 + (2.82)

X = 7.72 years

So, 3% of the time, the lifespan of the items will be above 7.72 years when 7 items are picked at random.