The fuel consumption, in miles per gallon, of all cars of a...

The fuel consumption, in miles per gallon, of all cars of a particular model has a mean of 25 and a standard deviation of 2. The population distribution can be assumed to be normal. A random sample of these cars is taken. a. Find the probability that sample mean fuel consumption will be fewer than 24 miles per gallon if i. a sample of 1 observation is taken. ii. a sample of 4 observations is taken. iii. a sample of 16 observations is taken. b. Explain why the three answers in part (a) differ in the way they do. Draw a graph to illustrate your reasoning.

The fuel consumption, in miles per gallon, of all cars of a particular model has a mean of 25 and a standard deviation of 2. The population distribution can be assumed to be normal. A random sample of these cars is taken.
a. Find the probability that sample mean fuel consumption will be fewer than 24 miles per gallon if i. a sample of 1 observation is taken.
ii. a sample of 4 observations is taken.
iii. a sample of 16 observations is taken.
b. Explain why the three answers in part (a) differ in the way they do. Draw a graph to illustrate your reasoning.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the probability distribution of the means of all possible samples of a fixed size drawn from a population. When sampling from a normally distributed population, the sample means themselves are normally distributed. This concept is crucial because it allows statisticians to make inferences about the population mean based on the sample mean, which may vary depending on the sample size.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve, defined by its mean and standard deviation. It is fundamental in statistics due to properties like symmetry about the mean and the fact that many natural phenomena approximate it. In this context, the fuel consumption follows a normal distribution, making calculations and probability estimations straightforward.

Standard Error

The standard error is a measure of the variability of the sample mean from sample to sample. It is calculated as the population standard deviation divided by the square root of the sample size. This concept explains why larger samples tend to produce sample means that are closer to the population mean, as the variability (or error) decreases with increasing sample size.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large. In scenarios where the population is already normally distributed, as in this case, the sample mean is exactly normally distributed. This theorem underpins many statistical procedures and explains how increased sample size leads to more precise estimates of the population mean.

Z-Score Transformation

The z-score transformation standardizes a value by subtracting the mean and dividing by the standard deviation of the distribution. This procedure converts values into a standard normal distribution, making it easier to calculate probabilities. In the context of the sample mean, z-scores help determine the probability of observing a sample mean less than a certain value by accounting for the effect of both the mean and the standard error.

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