The t-score is a statistical measure that represents the number of standard deviations a data point is from the mean of a population. It is used when the population standard deviation is unknown, and the sample size is small. The t-score is central to understanding various statistical concepts, including hypothesis testing and confidence intervals.
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The t-score is used when the population standard deviation is unknown, and the sample size is small (typically less than 30).
The t-score follows the Student's t-distribution, which has heavier tails than the normal distribution, reflecting the increased uncertainty when the population standard deviation is unknown.
The t-score is calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error of the mean.
The degrees of freedom for the t-score are equal to the sample size minus 1.
The t-score is used in hypothesis testing and the calculation of confidence intervals for a single population mean.
Review Questions
Explain how the t-score is used in the context of a single population mean using the Student's t-distribution (Section 8.2).
In the context of a single population mean using the Student's t-distribution (Section 8.2), the t-score is used to determine the statistical significance of the difference between the sample mean and the hypothesized population mean. The t-score is calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error of the mean. The resulting t-score is then compared to the critical value from the Student's t-distribution, which depends on the degrees of freedom (sample size minus 1) and the desired level of significance. If the t-score is greater than the critical value, the null hypothesis can be rejected, indicating that the difference between the sample mean and the hypothesized population mean is statistically significant.
Describe how the t-score is used in the calculation of confidence intervals for home costs (Section 8.4).
In the context of calculating confidence intervals for home costs (Section 8.4), the t-score is used to determine the range of values that are likely to contain the true population mean. The t-score is calculated using the sample mean, sample standard deviation, and sample size. The t-score is then used to determine the margin of error, which is added and subtracted from the sample mean to create the confidence interval. The level of confidence (e.g., 95%) determines the critical value from the Student's t-distribution that is used in the calculation. The resulting confidence interval represents the range of values that are likely to contain the true population mean of home costs with the specified level of confidence.
Analyze the role of the t-score in the context of hypothesis testing, including the sample, decision, and conclusion (Section 9.4).
In the context of hypothesis testing, including the sample, decision, and conclusion (Section 9.4), the t-score plays a crucial role. The t-score is used to determine the statistical significance of the difference between the sample statistic (e.g., sample mean) and the hypothesized population parameter. The t-score is calculated and compared to the critical value from the Student's t-distribution, which depends on the degrees of freedom and the desired level of significance. If the t-score is greater than the critical value, the null hypothesis can be rejected, and the conclusion can be drawn that the difference between the sample statistic and the hypothesized population parameter is statistically significant. The t-score is central to the decision-making process in hypothesis testing, as it provides the evidence needed to either support or reject the null hypothesis.
A probability distribution used when the population standard deviation is unknown, and the sample size is small. It is similar to the normal distribution but has heavier tails, reflecting the increased uncertainty when the population standard deviation is unknown.
The number of values in the final calculation of a statistic that are free to vary. In the context of the t-score, the degrees of freedom are equal to the sample size minus 1.