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 simple random sample of size 100 is selected and $x$ 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 simple random sample of size 100 is selected and $x$ 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

Central Limit Theorem

The central limit theorem states that as the sample size increases, the distribution of the sample mean tends to follow a normal distribution regardless of the underlying population’s distribution. This principle is particularly useful for large samples, allowing us to use normal probability calculations to estimate the likelihood of deviations of the sample mean from the population mean.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the probability distribution of all possible sample means drawn from a population. It is centered at the population mean and its spread, or variability, decreases as the sample size increases, which ensures that the sample mean becomes a more accurate estimator of the population mean.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This measure indicates how much the sample mean is expected to vary from the true population mean, thereby quantifying the precision of the estimate.

Z-Score and Its Application

A Z-score measures the number of standard deviations an observation is away from the mean. In the context of the sampling distribution, converting a range of values into Z-scores allows us to use the standard normal distribution to determine the probability that the sample mean falls within a certain range around the population mean.

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