The t-statistic is a statistical measure used to determine the significance of a sample mean or proportion compared to a hypothesized value. It is a crucial concept in hypothesis testing, as it helps assess whether the observed difference between a sample and a population is likely due to chance or represents a true difference.
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The t-statistic is used in hypothesis testing of a single mean (Chapter 9.6) and in hypothesis testing for two means (Chapter 10.5).
For a single mean, the t-statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean.
In the case of matched or paired samples (Chapter 10.4), the t-statistic is calculated using the mean of the differences between paired observations and the standard error of the mean difference.
The t-statistic follows a t-distribution, which is similar to the standard normal distribution but has heavier tails, accounting for the uncertainty in estimating the population standard deviation from a sample.
The t-statistic is compared to a critical value from the t-distribution to determine the statistical significance of the observed difference and make a decision about the null hypothesis.
Review Questions
Explain how the t-statistic is used in hypothesis testing of a single mean.
In the context of hypothesis testing of a single mean (Chapter 9.6), the t-statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean. This t-statistic is then compared to a critical value from the t-distribution to determine whether the observed difference is statistically significant, allowing the researcher to make a decision about the null hypothesis. The t-statistic takes into account the sample size and the variability in the sample, providing a measure of the strength of the evidence against the null hypothesis.
Describe the role of the t-statistic in hypothesis testing for two means.
When conducting hypothesis testing for two means (Chapter 10.5), the t-statistic is used to compare the difference between the two sample means to the expected difference under the null hypothesis. The t-statistic is calculated as the difference between the two sample means divided by the standard error of the difference. This t-statistic is then compared to a critical value from the t-distribution to determine the statistical significance of the observed difference and make a decision about the null hypothesis. The t-statistic in this context takes into account the variability within each sample and the sample sizes, providing a measure of the strength of the evidence for a difference between the two population means.
Analyze the use of the t-statistic in the context of matched or paired samples.
In the case of matched or paired samples (Chapter 10.4), the t-statistic is calculated using the mean of the differences between paired observations and the standard error of the mean difference. This approach accounts for the correlation between the paired observations, which can increase the statistical power of the test. The t-statistic in this context reflects the strength of the evidence for a difference between the paired observations, taking into consideration the variability in the differences and the sample size. The t-statistic is then compared to a critical value from the t-distribution to determine the statistical significance of the observed difference and make a decision about the null hypothesis.