Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.2 years, and standard deviation of 3 years. If you randomly purchase 24 items, what is the probability that their mean life will be longer than 14 years?

Answer 1

To find the probability that the mean lifespan of 24 items will be longer than 14 years, we use the Central Limit Theorem. The probability is approximately 0.0951.

Step-by-step explanation:

To find the probability that the mean lifespan of 24 items will be longer than 14 years, we can use the Central Limit Theorem. The Central Limit Theorem states that if a sample size is large enough, the sampling distribution of the mean will be approximately normally distributed regardless of the shape of the population distribution.

First, we find the distribution of the sample mean using the given population mean (13.2 years) and standard deviation (3 years). The standard deviation of the sample mean is the population standard deviation divided by the square root of the sample size. So, the standard deviation of the sample mean is 3 / sqrt(24) = 0.61237 years.

Next, we calculate the z-score for a mean lifespan of 14 years using the formula: z = (x - μ) / σ, where x is the desired mean lifespan, μ is the population mean, and σ is the standard deviation of the sample mean. Plugging in the values, we get z = (14 - 13.2) / 0.61237 = 1.3047.

Finally, we use the z-score to find the probability using a standard normal distribution table or a calculator. The probability that the mean lifespan will be longer than 14 years is the probability to the right of the z-score, which is 1 - P(z < 1.3047). Using a standard normal distribution table or a calculator, we find this probability to be approximately 0.0951.