t^* is a test statistic used in statistical inference, specifically in the context of estimating a single population mean using the Student's t-distribution. It represents the standardized difference between the sample mean and the hypothesized population mean, taking into account the sample size and the standard deviation of the sample.
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The t^* statistic follows a Student's t-distribution with n-1 degrees of freedom, where n is the sample size.
t^* is used to determine whether the sample mean is significantly different from the hypothesized population mean, based on the level of significance chosen for the test.
The value of t^* is calculated as (sample mean - hypothesized population mean) / (standard error of the sample mean).
The t^* statistic is compared to a critical value from the Student's t-distribution to determine whether to reject or fail to reject the null hypothesis.
The t^* statistic is also used to construct a confidence interval for the population mean, which provides a range of values that are likely to contain the true population mean.
Review Questions
Explain the role of the t^* statistic in the context of estimating a single population mean using the Student's t-distribution.
The t^* statistic is a key component in the process of estimating a single population mean using the Student's t-distribution. It represents the standardized difference between the sample mean and the hypothesized population mean, taking into account the sample size and the standard deviation of the sample. This statistic follows a Student's t-distribution with n-1 degrees of freedom, where n is the sample size. The value of t^* is used to determine whether the sample mean is significantly different from the hypothesized population mean, based on the chosen level of significance. Additionally, the t^* statistic is used to construct a confidence interval for the population mean, providing a range of values that are likely to contain the true population mean.
Describe how the t^* statistic is calculated and how it is used in hypothesis testing for a single population mean.
The t^* statistic is calculated as (sample mean - hypothesized population mean) / (standard error of the sample mean). This standardized difference takes into account the variability in the sample and the sample size. In the context of hypothesis testing for a single population mean, the t^* statistic is compared to a critical value from the Student's t-distribution, which is determined by the chosen level of significance and the degrees of freedom (n-1). If the absolute value of t^* is greater than the critical value, the null hypothesis (that the population mean is equal to the hypothesized value) is rejected, indicating that the sample mean is significantly different from the hypothesized population mean. Conversely, if the absolute value of t^* is less than the critical value, the null hypothesis is not rejected, suggesting that the sample mean is not significantly different from the hypothesized population mean.
Explain how the t^* statistic is used to construct a confidence interval for the population mean and discuss the interpretation of the confidence interval.
The t^* statistic is used to construct a confidence interval for the population mean by incorporating the sample mean, the standard error of the sample mean, and a critical value from the Student's t-distribution. The formula for the confidence interval is: sample mean ± (t^* × standard error of the sample mean). The confidence level, typically 95%, determines the critical value used in the calculation. The interpretation of the confidence interval is that if the sampling process were repeated many times, the population mean would be contained within the calculated interval in a certain proportion of the cases (e.g., 95% of the time for a 95% confidence interval). This provides a range of plausible values for the true population mean, taking into account the uncertainty associated with the sample. The width of the confidence interval is influenced by the sample size, the variability in the sample, and the desired level of confidence, with larger samples and lower variability leading to narrower confidence intervals.
A probability distribution used to make inferences about the mean of a population when the sample size is small, and the population standard deviation is unknown.