Final answer:
The question asks about the sampling distribution of a sample mean from a normal population. It provides information about car battery lifetimes and asks for the mean and standard deviation of the sampling distribution for a sample size of 8, the interpretation of the standard deviation, and the probability that the sample mean life is less than a certain value.
Step-by-step explanation:
The question is asking about the sampling distribution of a sample mean, specifically when sampling from a normal population. In this case, the question provides information about the lifetime of car batteries from a manufacturer, which follows a normal distribution with a mean of 48 months and a standard deviation of 8.2 months.
To calculate the mean and standard deviation of the sampling distribution for a sample size of 8, we use the formulas:
Mean of sampling distribution: μ_x = μ
Standard deviation of sampling distribution: σ_x = σ / sqrt(n)
Plugging in the values, we have:
μ_x = 48 months
σ_x = 8.2 months / sqrt(8)
Interpreting the standard deviation, σ_x, it represents the average amount of variability in the sample means from different samples of size 8. A smaller standard deviation indicates less variability and a more precise estimate of the true population mean.
To find the probability that the sample mean life is less than 42.2 months, we need to calculate the z-score and use a standard normal distribution table. The formula for the z-score is:
z = (x - μ) / (σ / sqrt(n))