# Activity 10/01/2021

## STAT 445 / 645 -- Section 1001 <br> Instructor: Colin Grudzien<br>

## Instructions:
We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms.  Follow the instructions in each sub-section when the instructor assigns you to a breakout room.

## Activities:


### Activity: Computing t-confidence intervals manually versus automatically


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 1:

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 2:

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.


### Activity: General hypothesis testing, power of tests and sample sizes

### Question 1:

Cloud seeding has been studied for many decades as a weather modification procedure (for an interesting study of this subject, see the article in Technometrics, "A Bayesian Analysis of a Multiplicative Treatment Effect in Weather Modification", Vol. 17, pp. 161–166). The rainfall in acre-feet from 20 clouds that were selected at random and seeded with silver nitrate follows:

```{r}
rainfall <- c(18.0, 30.7, 19.8, 27.1, 22.3, 18.8, 31.8, 23.4, 21.2, 27.9, 31.9, 27.1, 25.0, 24.7, 26.9, 21.8, 29.2, 34.8, 26.7, 31.6)
```

Can you support a claim that mean rainfall from seeded clouds exceeds 25 acre-feet? Use $\alpha = 0.01$ significance.



### Question 2:

Compute the power of the test if the true mean rainfall is 27 acre-feet.


#### Question 3:

What sample size would be required to detect a true mean rainfall of 27.5 acre-feet if we wanted the power of the test to be at least 0.9?

