## Student Name:
## Instructions:
Fill in your name above. We will work through the following activities as a group in the discussion sections. You must turn in a completed worksheet with solutions filled in below for credit. The solutions will be given in the discussion section.
## Activity: Computing t-critical values
### Question 1:
Let the t critical value, $t_\frac{\alpha}{2}$ for $n-1=15$ degrees of freedom be denoted $t_{0.025,15}$. Use the `qt` function to determine the value of $t_{0.025,15}$.
### Question 2:
Let $t_\frac{\alpha}{2}$ be the critical value for the two-sided confidence interval
$$\left(\overline{x} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}, \overline{x} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\right)$$
corresponding to a confidence level of $90\%$, and $n-1$ degrees of freedom$=25$
Find the appropriate $t_\frac{\alpha}{2}$.
### Question 3:
Let $t_\alpha$ be the critical value for the one-sided confidence bound
$$\left(-\infty, \overline{x} + \hat{\sigma}_\overline{X} t_\alpha\right)$$
corresponding to a confidence level of $99\%$, and $n-1$ degrees of freedom$=19$
Find the appropriate $t_\alpha$.
## Activity: Computing t-confidence intervals
An article in Computers & Electrical Engineering, “Parallel simulation of cellular neural networks” (1996, Vol. 22, pp. 61–84) considered the speed-up of cellular neural networks (CNN) for a parallel general-purpose computing architecture based on six transputers in different areas. The data follows:
```{r}
speed_up_times <- c(3.775302, 3.350679, 4.217981, 4.030324, 4.639692, 4.139665, 4.395575, 4.824257, 4.268119, 4.584193, 4.930027, 4.315973, 4.600101)
```
Assume population is approximately normally distributed.
### Question 4:
Compute a two-sided t-confidence interval for the population mean with a $95\%$ level of confidence manually. Then use the `t.test()` function to verify that this was computed correctly.
### Question 5:
Compute a one-sided, upper t-confidence bound on the population mean with a $99\%$ level of confidence manually. Then use the `t.test()` function to verify that this was computed correctly.