Sample Means

Definition

The sample mean is the arithmetic average of a set of observations drawn from a population. It represents the central tendency of the sample and is a crucial statistic used to make inferences about the population from which the sample was drawn.

5 Must Know Facts For Your Next Test

1. The sample mean is an unbiased estimator of the population mean, meaning that on average, the sample mean will equal the population mean.
2. As the sample size increases, the sampling distribution of the sample mean becomes more tightly clustered around the population mean, with a smaller standard deviation.
3. The Central Limit Theorem allows us to use the normal distribution to make inferences about the population mean, even if the population distribution is not normal.
4. The standard error of the sample mean is the standard deviation of the sampling distribution of the sample mean, and it decreases as the sample size increases.
5. The sample mean is a key statistic used in hypothesis testing and confidence interval estimation to make inferences about the population.

Review Questions

• Explain how the sample mean is used to make inferences about the population mean.
  • The sample mean is an unbiased estimator of the population mean, meaning that on average, the sample mean will equal the population mean. As the sample size increases, the sampling distribution of the sample mean becomes more tightly clustered around the population mean, with a smaller standard deviation. This allows us to use the normal distribution to make inferences about the population mean, even if the population distribution is not normal. For example, we can use the sample mean to construct a confidence interval for the population mean or to test hypotheses about the population mean.
• Describe the relationship between the sample size and the sampling distribution of the sample mean.
  • According to the Central Limit Theorem, as the sample size increases, the sampling distribution of the sample mean becomes more tightly clustered around the population mean, with a smaller standard deviation. This means that as the sample size increases, the sample mean becomes a more precise estimate of the population mean. The standard error of the sample mean, which is the standard deviation of the sampling distribution of the sample mean, decreases as the sample size increases. This allows us to make more accurate inferences about the population mean as the sample size grows.
• Evaluate the importance of the sample mean in the context of hypothesis testing and confidence interval estimation.
  • The sample mean is a crucial statistic in both hypothesis testing and confidence interval estimation. In hypothesis testing, the sample mean is used to test hypotheses about the population mean, such as whether the population mean is equal to a specific value or whether it is different from another population mean. The sample mean is also used to construct confidence intervals for the population mean, which provide a range of plausible values for the true population mean based on the sample data. These statistical inferences made using the sample mean are essential for drawing conclusions about the population and making informed decisions in various fields, such as business, economics, and social sciences.

Related terms

Population Mean: The population mean is the arithmetic average of all the values in a given population. It is the true central tendency of the entire population.

Central Limit Theorem: The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution.

Sampling Distribution: The sampling distribution is the probability distribution of a statistic, such as the sample mean, across all possible samples of the same size drawn from a population.