**Table C: t Distribution Critical Values** This table provides critical values for the t distribution, which are crucial for conducting hypothesis tests and constructing confidence intervals. To compare P-values with t statistics, match the critical values of t* with the P-values given at the bottom of the table. **Chart Explanation** The chart above the table illustrates a normal distribution curve, highlighting "Area C" in the center and "Tail area" represented by C/2 on each end. The markings "-t*" and "t*" indicate the critical points that define the boundaries of the confidence interval. ### Table Columns 1. **Degrees of Freedom (DF)**: Listed vertically, ranging from 1 to 1000, which is essential for determining the correct t value. 2. **Confidence Level C**: Runs horizontally with percentages from 50% to 99.9%. These levels indicate the probability that the confidence interval contains the true parameter. ### Table Rows Each row corresponds to degrees of freedom and contains critical t values for various confidence levels: - **50%**: Critical values from 1.000 to 0.675 - **60%**: Critical values from 1.376 to 0.842 - **70%**: Critical values from 1.963 to 1.292 - **80%**: Critical values from 3.078 to 1.282 - **90%**: Critical values from 6.314 to 1.646 - **95%**: Critical values from 12.71 to 1.962 - **96%**: Critical values from 15.89 to 2.054 - **98%**: Critical values from 31.82 to 2.326 - **99%**: Critical values from 63.66 to 2.576 - **99.5%**: Critical values from 127.3 to 2.807 - **99.8%**: Critical values from 318.3 to 3.291 - **99.9%**: Critical values from 636.6 to 3.807 ### One-sided and Two-sided P-values - **One-sided P**: Represents tail probabilities for one side, ranging from 0.25 to 0.0005. - **Two-sided P**: Represents probabilities for both tails, ranging from 0.50 We randomly select 10 individuals from a large group who have participated in an SAT-Math tutorial. We would like to find out if there is good evidence that the tutorial improves average scores on the SAT-M Math test. Each individual's baseline score and score after the tutorial is recorded below (along with the difference between the baseline and the after tutorial test). We can assume that the distribution of differences is relatively normal. | Individual # | SAT-M Baseline | SAT-M (after tutorial) | Difference | |--------------|----------------|------------------------|------------| | 1 | 680 | 690 | 10 | | 2 | 540 | 560 | 20 | | 3 | 590 | 600 | 10 | | 4 | 620 | 620 | 0 | | 5 | 630 | 600 | -30 | | 6 | 660 | 670 | 10 | | 7 | 490 | 500 | 10 | | 8 | 510 | 500 | -10 | | 9 | 700 | 710 | 10 | | 10 | 420 | 443 | 23 | **The average difference from the sample is 5.3 and the sample standard deviation is 15.4.** A researcher hopes to find evidence that the tutorial **increases** overall SAT-M scores. 1. **What are the null and alternative hypothesis statements for this problem?** 2. **Using the equation below, calculate the resulting t-statistic (show your work and/or describe the process)** \[ t = \frac{\bar{x} - 0}{s/\sqrt{n}} \] 3. **Report the degrees of freedom and the critical value for an alpha level of .05 (see table below)** 4. **Can we reject the null hypothesis? Why or why not?**