σx̄

Definition

σx̄ represents the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. It is a measure of the variability or spread of the sample means that would be obtained from repeated sampling of the population.

5 Must Know Facts For Your Next Test

1. The value of σx̄ decreases as the sample size (n) increases, indicating that the sample means become less variable as the sample size gets larger.
2. The formula for σx̄ is σx̄ = σ/√n, where σ is the standard deviation of the population and n is the sample size.
3. σx̄ is used to construct confidence intervals and conduct hypothesis tests about the population mean, μ.
4. The Central Limit Theorem ensures that the sampling distribution of the sample mean is approximately normal, even if the original population distribution is not normal, as long as the sample size is sufficiently large.
5. The standard error of the mean, σx̄, is a crucial concept in statistical inference, as it allows us to quantify the uncertainty associated with sample estimates of the population mean.

Review Questions

• Explain the relationship between the sample size (n) and the value of σx̄.
  • As the sample size (n) increases, the value of σx̄ decreases. This is because the standard error of the mean, σx̄, is inversely proportional to the square root of the sample size (σx̄ = σ/√n). This means that as the sample size gets larger, the sample means become less variable, and the standard error of the mean gets smaller. This is a key implication of the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.
• Describe how σx̄ is used in the construction of confidence intervals for the population mean, μ.
  • The standard error of the mean, σx̄, is a crucial component in the construction of confidence intervals for the population mean, μ. The formula for a confidence interval for the population mean is: $\bar{x} \pm z_{\alpha/2} \cdot \sigma_{\bar{x}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the critical value from the standard normal distribution, and $\sigma_{\bar{x}}$ is the standard error of the mean, σx̄. The width of the confidence interval is directly proportional to the value of σx̄, so as σx̄ decreases (with larger sample sizes), the confidence interval becomes narrower, indicating more precision in the estimate of the population mean.
• Explain how the Central Limit Theorem and the concept of σx̄ are used to conduct hypothesis tests about the population mean, μ.
  • The Central Limit Theorem and the standard error of the mean, σx̄, are essential for conducting hypothesis tests about the population mean, μ. The test statistic used in these tests is the $z$-score, which is calculated as: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the hypothesized population mean, and $\sigma_{\bar{x}}$ is the standard error of the mean, σx̄. The Central Limit Theorem ensures that the sampling distribution of the sample mean is approximately normal, allowing us to use the standard normal distribution to determine the $p$-value and make inferences about the population mean. The value of σx̄ directly affects the precision and power of these hypothesis tests.

Related terms

Central Limit Theorem:

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

Standard Deviation:

Standard deviation is a measure of the spread or variability of a set of data values around the mean. It represents the average amount of deviation from the mean.

Sampling Distribution:

The sampling distribution is the probability distribution of a statistic, such as the sample mean, obtained by taking all possible samples of the same size from a population.