T-statistic

Definition

The t-statistic is a statistical measure used to determine the probability that the difference between two sample means is due to chance. It is commonly used in hypothesis testing and confidence interval estimation when the population standard deviation is unknown and the sample size is small.

5 Must Know Facts For Your Next Test

1. The t-statistic follows a t-distribution, which is a bell-shaped curve that is more spread out than the normal distribution.
2. The t-statistic is used when the population standard deviation is unknown and the sample size is small (typically less than 30).
3. The t-statistic is used to construct confidence intervals for the population mean when the population standard deviation is unknown.
4. The t-statistic is also used to compare the means of two independent populations when the population standard deviations are unknown.
5. The t-statistic is used to analyze matched or paired samples, where the observations in the two samples are related or paired in some way.

Review Questions

• Explain how the t-statistic is used to construct a confidence interval for the population mean when the population standard deviation is unknown.
  • When the population standard deviation is unknown, the t-statistic is used to construct a confidence interval for the population mean. The formula for the confidence interval is: $\bar{x} \pm t_{\alpha/2,n-1} \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2,n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom. The t-statistic is used to determine the critical value, which accounts for the uncertainty in the population standard deviation.
• Describe how the t-statistic is used to compare the means of two independent populations when the population standard deviations are unknown.
  • When comparing the means of two independent populations and the population standard deviations are unknown, the t-statistic is used to determine if the difference between the means is statistically significant. The formula for the t-statistic is: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$, where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $s_1^2$ and $s_2^2$ are the sample variances, and $n_1$ and $n_2$ are the sample sizes. The calculated t-statistic is then compared to a critical value from the t-distribution to determine if the difference between the means is statistically significant.
• Analyze how the t-statistic is used in the context of matched or paired samples, where the observations in the two samples are related or paired in some way.
  • In the case of matched or paired samples, where the observations in the two samples are related or paired in some way, the t-statistic is used to determine if the difference between the paired observations is statistically significant. The formula for the t-statistic in this case is: $t = \frac{\bar{d}}{s_d/\sqrt{n}}$, where $\bar{d}$ is the mean of the differences between the paired observations, $s_d$ is the standard deviation of the differences, and $n$ is the number of paired observations. The calculated t-statistic is then compared to a critical value from the t-distribution to determine if the difference between the paired observations is statistically significant. This approach is useful when the observations in the two samples are not independent, such as in before-and-after studies or when measuring the same individuals under different conditions.

Related terms

Hypothesis Testing: A statistical method used to determine whether a claim or hypothesis about a population parameter is likely to be true or false.

Confidence Interval: A range of values that is likely to contain an unknown population parameter with a specified level of confidence.

Sample Size: The number of observations or data points collected from a population for the purpose of statistical analysis.