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Chapter 9: Problem 22

Which distribution do you use when you are testing a population mean and the standard deviation is known? Assume sample size is large.

Short Answer

Expert verified
Use the normal distribution when the population standard deviation is known and the sample size is large.

Step by step solution

01

Understanding the Problem

When we are testing a population mean, we need to choose the appropriate distribution based on the information available. Given that the standard deviation is known and the sample size is large, we must identify the right distribution to use.
02

Recalling the Large Sample Distribution

For large samples, where the sample size typically exceeds 30, the Central Limit Theorem suggests that the sampling distribution of the sample mean is approximately normally distributed, regardless of the shape of the population distribution.
03

Standard Deviation Known

When the population standard deviation is known, it is not necessary to estimate it from the sample data. This allows us to use the normal distribution for hypothesis testing of the population mean, rather than the t-distribution.
04

Choice of Distribution

With a known population standard deviation and a large sample size, the normal distribution, specifically the standard normal distribution (z-distribution), is appropriate for conducting hypothesis tests on the population mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean
The term 'population mean' refers to the average of all values in a population. It is symbolized by the Greek letter \( \mu \). Understanding the population mean is crucial since it provides a central location measure of the entire population.
  • Calculating the population mean involves summing all the values in the population and then dividing by the total number of values.
  • The formula is \( \mu = \frac{\sum X}{N} \), where \( \sum X \) is the sum of all population values and \( N \) is the number of values.
In statistical analysis, knowing the population mean helps in making predictions and checking assumptions against sample data.
It acts as a reference point for comparing sample means to determine the accuracy of our sample as a reflection of the population.
Standard Deviation
Standard deviation is a measure that quantifies the amount of variation or dispersion within a set of values.
  • In the context of a known standard deviation, it provides valuable information about how much individual data points differ from the mean of the data set.
  • The formula for standard deviation of a population is \( \sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}} \), where \( X \) represents individual data points, \( \mu \) is the population mean, and \( N \) is the number of observations.
When the standard deviation is known, it simplifies statistical analyses, such as hypothesis testing. For a large sample size where the data distribution is approximately normal, the standard deviation is crucial in determining how data spreads around the population mean, assisting in the accuracy of statistical inferences.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that the distribution of the sample means will approximate a normal distribution as the sample size becomes large, regardless of the population's original distribution.
  • This theorem is crucial because it underpins many statistical procedures, making them applicable even when population distributions are not normal.
  • Most importantly, the CLT implies that the sample mean will also approach the population mean given a sufficiently large sample size.
In practical terms, the CLT allows us to conduct hypothesis tests using the normal distribution, even if the underlying population distribution is unknown or non-normal, provided the sample size is large enough (typically greater than 30). This characteristic greatly simplifies many aspects of statistical analysis and helps statisticians make inferences about population parameters.

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