A population has mean 1,542 and standard deviation 246 . a....

A population has mean 1,542 and standard deviation $246 .$
a. Find the mean and standard deviation of $x-$ for samples of size 100 .
b. Find the probability that the mean of a sample of size 100 will be within 100 units of the population mean, that is, between 1,442 and 1,642 .

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Key Concepts

Standardization (Z-score Calculation)

This concept involves converting a value from a normal distribution with any mean and standard deviation into a standard normal distribution (mean 0, standard deviation 1). This allows for the utilization of standard normal tables to compute probabilities associated with specific intervals.

Standard Error

Standard error refers to the standard deviation of the sampling distribution of the sample mean. It quantifies how much the sample mean is expected to vary from one sample to another and is computed by dividing the population standard deviation by the square root of the sample size.

Sampling Distribution of the Sample Mean

This concept describes the probability distribution of all possible sample means of a specified size drawn from the population. Its mean is equal to the population mean, and its standard deviation (often referred to as the standard error) is equal to the population standard deviation divided by the square root of the sample size.

Central Limit Theorem

The central limit theorem states that, regardless of the population's distribution, the distribution of the sample mean will tend to be approximately normal if the sample size is sufficiently large. This result justifies the use of normal probabilities to estimate probabilities related to the sample mean.

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