A population has a mean of 200 and a standard deviation of...

A population has a mean of 200 and a standard deviation of 50 . Suppose a sample of size 100 is selected and $\bar{x}$ is used to estimate $\mu$. a. What is the probability that the sample mean will be within $\pm 5$ of the population mean? b. What is the probability that the sample mean will be within $\pm 10$ of the population mean?

A population has a mean of 200 and a standard deviation of 50 . Suppose a sample of size 100 is selected and $\bar{x}$ is used to estimate $\mu$.
a. What is the probability that the sample mean will be within $\pm 5$ of the population mean?
b. What is the probability that the sample mean will be within $\pm 10$ of the population mean?

Key Concepts

Z-Score and the Standard Normal Distribution

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It is used to convert a sample mean to a common scale which enables one to calculate probabilities using the standard normal distribution.

Probability Calculation for Sample Means

To determine the probability that the sample mean falls within a specific range around the population mean, the Z-scores corresponding to the endpoints of the range are computed using the standard error. Then, the probability is found by referring to the cumulative distribution function of the standard normal distribution, yielding the desired probability for the sample mean.

Central Limit Theorem

This theorem states that the distribution of the sample means approximates a normal distribution, regardless of the population’s distribution, as long as the sample size is sufficiently large. It justifies using the normal model for problems involving the sample mean, especially when dealing with sums or averages of random variables.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated as the population standard deviation divided by the square root of the sample size. In this context, it quantifies the dispersion of sample means around the true population mean.

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