Procter & Gamble reported that an American family of four...

Procter $\&$ Gamble reported that an American family of four washes an average of 1 ton ( 2000 pounds) of clothes each year. If the standard deviation of the distribution is 187.5 pounds, find the probability that the mean of a randomly selected sample of 50 families of four will be between 1980 and 1990 pounds.

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a statistical principle that states when a large enough sample is taken from any population with a finite level of variance, the distribution of the sample means will be approximately normally distributed. This holds regardless of the shape of the population distribution, which provides the foundation for making inferences about the population mean using the sample mean in problems involving probability and statistical estimation.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the probability distribution of all possible sample means of a given size taken from the same population. In this context, it allows us to calculate probabilities concerning the average weight of clothes washed by families, by considering the distribution of the sample mean rather than individual family values.

Standard Error

The standard error measures the dispersion of the sample means around the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. The standard error quantifies the uncertainty associated with estimating the population mean from a sample, and it is crucial for constructing probabilities and confidence intervals in statistics.

Z-Scores and the Standard Normal Distribution

Z-scores are standardized values that measure the number of standard errors a sample mean is away from the population mean. By transforming sample means into Z-scores, statisticians can use the standard normal distribution to determine probabilities. This process is utilized to find the probability that the sample mean falls within a specified range by referencing standard normal probability tables or computational tools.

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