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The standard error is a vital concept when dealing with probability distributions in statistics. It's a measure that tells us how much the sample mean of a distribution is likely to vary from the true population mean. In simpler terms, it helps us understand how close our sample mean is to the population mean.
For any given sample size (), the standard error of the mean (SE) is calculated with the formula:

• SE = \( \frac{\sigma}{\sqrt{n}} \)

where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
The standard error decreases as the sample size increases because larger samples tend to better approximate the population mean. In our example, the standard deviation is 187.5 pounds and the sample size is 50 families. The calculation of SE provides a foundation for further steps, particularly in the calculation of Z-scores.

Z-scores are a way of standardizing scores so they can be compared across different data sets. They indicate how many standard deviations a specific point is from the mean.
For our problem, we calculate Z-scores for the sample mean lower and upper bounds: 1980 and 1990 pounds.
The formula for calculating a Z-score is:

• \( Z = \frac{\bar{x} - \mu}{SE} \)

where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, and SE is the standard error calculated earlier.
Using these calculations, we can find:

• For 1980: \( Z_{1980} = \frac{1980 - 2000}{SE} \)
• For 1990: \( Z_{1990} = \frac{1990 - 2000}{SE} \)

Z-scores help in determining how likely each of these values is, relative to the normal distribution.

The cumulative distribution function (CDF) is a fundamental concept in probability that shows the probability that a random variable is less than or equal to a certain value.
When you have a Z-score, you can use a Z-table to find its corresponding CDF value. For any value \( x \), the CDF gives you the area under the curve to the left of \( x \) in a probability distribution.
In the given exercise, we are concerned with the CDF values corresponding to the Z-scores we calculated earlier:

• Find \( P(Z_{1980}) \) using the Z-table
• Find \( P(Z_{1990}) \) using the Z-table

The difference between these CDF values will provide the probability that the sample mean falls between our specified limits.

The population mean \( \mu \) is the average of all possible values in a population. It is a key parameter in the computation of probabilities and the comprehension of data distributions.
In our exercise, the population mean is given as 2000 pounds. This number represents the expected average amount of laundry washed by a family of four annually.
Understanding the population mean allows us to compare how much our sample mean deviates from this central figure. It also helps in calculating Z-scores, which relate the population mean to the specific points of interest in our sample data.
Moreover, it's crucial for understanding the effectiveness of the sample we have collected in representing the whole population.