SOLUTION: A manufacturer knows that their items have a normally distributed lifespan, with a mean of 14.4 years, and standard deviation of 4.7 years. If 14 items are picked at random, 2%

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Question 1184574: A manufacturer knows that their items have a normally distributed lifespan, with a mean of 14.4 years, and standard deviation of 4.7 years.
If 14 items are picked at random, 2% of the time their mean life will be less than how many years?
Give your answer to one decimal place.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
population mean is 14.4 years.
population standard deviation is 4.7 years.
sample size is 14.
degrees of freedom = 13

based on the guideline at https://math.stackexchange.com/questions/1817980/how-to-know-when-to-use-t-value-or-z-value, you would use the t-score for this problem.

standard error = standard deviation / square root of sample size = 4.7 / sqrt(14) = 1.25613.

t-score with 2% of the area under the normal distribution curve to the left of it with 13 degrees of freedom is -2.2816.

use the t-score formula to find the raw score.

t-score formula is t = (x - m) / s

that becomes:

-2.2816 = (x - 14.4) / 1.25613.

solve for x to get:

x = 1.25613 * -2.2816 + 14.4 = 11.534.

round to 1 decimal place to get 11.5.

that's your answer.