A manufacturer produces 25-pound lifting weights. The lowest...

A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken. a. What is the distribution for the weights of one 25 -pound lifting weight? What is the mean and standard deivation? b. What is the distribution for the mean weight of 10025 -pound lifting weights? c. Find the probability that the mean actual weight for the 100 weights is less than $24.9 .$

A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken.
a. What is the distribution for the weights of one 25 -pound lifting weight? What is the mean and standard deivation?
b. What is the distribution for the mean weight of 10025 -pound lifting weights?
c. Find the probability that the mean actual weight for the 100 weights is less than $24.9 .$

Key Concepts

Uniform Distribution

A uniform distribution is a probability distribution in which all outcomes are equally likely within a specified interval. In this scenario, the weight of a lifting weight is equally likely to be any value between 24 and 26 pounds. The mean of a uniform distribution over the interval [a, b] is given by (a + b)/2 and the variance is computed as ((b - a)^2)/12, which allows for finding the standard deviation by taking the square root of the variance.

Sampling Distribution of the Mean

When taking a sample from a population, the distribution of the sample mean forms its own probability distribution. This distribution has a mean equal to the population mean and a variance equal to the population variance divided by the sample size (n). For large sample sizes, this distribution is approximately normal regardless of the original population distribution, enabling probabilistic statements about sample averages.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental statistical principle stating that, for large enough sample sizes, the distribution of the sample mean will approximate a normal distribution even if the underlying population distribution is not normal. This theorem justifies the use of normal probability calculations in problems involving the mean of multiple independent samples.

Standardization and Z-Score Calculation

Standardization is the process of converting a random variable to a standardized scale, typically by subtracting its mean and dividing by its standard deviation to obtain a Z-score. This transformation enables the use of the standard normal distribution for calculating probabilities. For the sample mean, the Z-score is computed using the difference between the observed mean and the population mean, divided by the standard error (which is the population standard deviation divided by the square root of the sample size).

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