The fuel economy of a 2009 Lexus RX 350 2WD 6 cylinder 3.5 L...

The fuel economy of a 2009 Lexus RX 350 2WD 6 cylinder 3.5 L automatic 5 -speed using premium fuel is a normally distributed random variable with a mean of $\mu=20.5 \mathrm{MPG}$ and a standard deviation of $\sigma=1.25 \mathrm{MPG}$. (a) What is the standard error of $\bar{X}$, the mean from a random sample of 16 fill-ups by one driver? (b) Within what interval would you expect the sample mean to fall, with 90 percent probability?

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Key Concepts

Standard Error

The standard error is a measure of the variability or dispersion of the sample mean around the true population mean. It quantifies how much the sample mean is expected to vary from sample to sample due to random sampling variability. Mathematically, it is calculated as the population standard deviation divided by the square root of the sample size.

Sampling Distribution of the Sample Mean

When samples are drawn from a population, the distribution of the sample means is known as the sampling distribution. If the original data are normally distributed, the distribution of the sample mean is also normal irrespective of the sample size. This is a central concept in inferential statistics, as it allows us to make probability statements about sample means.

Confidence Interval

A confidence interval provides a range within which a population parameter, such as the mean, is expected to lie with a certain level of confidence. In the context of the sample mean, it quantifies the uncertainty around the estimated mean and is often expressed in terms of a probability such as 90% or 95%. This interval is constructed based on the sampling distribution and standard error, incorporating the desired level of confidence.

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