Chapter9.Rmd

# Ch. 9 Inferences Based on Two Samples {-}

Sections covered: 9.1, 9.2, 9.3, 9.4

## 9.1 $z$ Tests and CI's for a Difference Between Two Population Means {-}

(Cases 1 & 2)

Skip: $\beta$ and the Choice of Sample Size (pp. 366-367)

### R {-}

Since you aren't given the original data, R isn't very helpful here, but you can write a function that you could reuse, such as:

```{r}
options(scipen = 999) # get rid of scientific notation

# this function only has to be run once per session, and then you can reuse it.

diffmeans <- function(xbar, ybar, delta0,
                      sigma1, sigma2,
                      m, n, type = "twosided") {
  
Z <- ((xbar - ybar) - delta0)/sqrt(((sigma1^2)/m) + ((sigma2^2)/n))


if (type == "twosided") {
  pvalue <- pnorm(-abs(Z))*2
} else if (type == "lowertail") {
  pvalue <- pnorm(Z)
} else {
  pvalue <- 1 - pnorm(Z)
}

print(c("The p-value is", round(pvalue, 4)))

}


# Example 9.1, p. 365

diffmeans(xbar  = 29.8, ybar  = 34.7,
          delta0 = 0, sigma1 = 4, sigma2 = 5,
          m = 20, n = 25, type = "twosided")

# Example 9.4, p. 368

# As long as you use the correct order of parameters, you don't have to write out the names:

diffmeans(2258, 2637, -200, 1519, 1138, 663, 413, "lowertail")
```

## 9.2 The Two-Sample $t$ Test and CI {-}

(Case 3)

Use: https://www.statology.org/welchs-t-test-calculator/ (choose "Enter raw data" or "Enter summary data" as appropriate) to calculate the degrees of freedom.

Skip: Pooled $t$ Procedures (pp. 377-378)

Skip: Type II Error Probabilities (pp. 378-379)

Given two random samples, use `t.test()` with different parameters to carry out a two-sample hypothesis test.

For demonstration purposes, x and y here are samples of 10 numbers that drawn from the standard normal distribution.

```{r}
set.seed(1)
x <- rnorm(10)
set.seed(2)
y <- rnorm(10)
t.test(x, y, var.equal = TRUE, alternative = "less") 
```
Thus, H_0 is rejected at alpha = .05.

In `t.test()`, "less" indicates that H_A: delta < 0. Also try using `alternative = "two.sided"` or `alternative = "two.sided"` for a different alternative hypothesis.


## 9.3 Analysis of Paired Data {-}

Skip: Paired Data and Two-Sample $t$ Procedures (pp. 386-387)

Skip: Paired Versus Unpaired Experiments (pp. 387-388)

Two ways of doing paired test:
(Here we continue to use the `x` and `y` in section 9.2 above)
**[Method 1]** Take x-y and do a one-sample test. 
```{r}
t.test(x-y, alternative = "less")
```

**[Method 2]** Give another parameter `paired = TRUE`. In R, the default parameter for `paired` in `t.test()` is `FALSE`; here we set it to `TRUE` and leave `x` and `y` as two separate inputs.
```{r}
t.test(x, y, alternative = "less", paired = TRUE)
```

It's clear that they give the same results.


## 9.4 Inferences Concerning a Difference Between Population Proportions {-}

Skip: Type II Error Probabilities and Sample Sizes (pp. 394-395)

### R {-}

A/B Testing question from class:

```{r}
clicks <- c(25, 20)
people <- c(100, 100)
prop.test(clicks, people, correct = FALSE)
```