A researcher reports survey results by stating that the...

A researcher reports survey results by stating that the standard error of the mean is $20 .$ The population standard deviation is 500 .
a. How large was the sample used in this survey?
b. What is the probability that the point estimate was within $\pm 25$ of the population mean?

Key Concepts

Standard Normal Distribution and Z-Scores

Z-scores standardize differences by expressing them in terms of standard deviations, allowing the use of the standard normal distribution to compute probabilities. This method is essential for determining the likelihood that a sample statistic falls within a specified range around a population parameter.

Central Limit Theorem

The central limit theorem underpins the use of the normal distribution in statistical inference by stating that, for a sufficiently large sample, the distribution of the sample mean tends toward a normal distribution regardless of the population's distribution. This justifies applying normal probability calculations when estimating the probability of the sample mean falling within a specific interval.

Standard Error

The standard error is a measure of the variability of the sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size, quantifying the precision of the estimate of the population mean.

Sample Size Determination

This concept involves using the relationship between the standard error, population standard deviation, and sample size to estimate how many observations are needed to achieve a designated level of precision for an estimate. By rearranging the standard error formula, one can solve for the sample size required.

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