---
title: "Lec 01 code"
format: html
---

## Example

Here is some data and the normal QQ plot:

```{r}
wt <- c(41,34,40,44,33,42,52,38,32,31,31,35,
39,44,42,42,35,33,40,48,51,32,41,35,
38,48,37,35,42,41,40,47,40,46,33,38,
51,39,40)
qqnorm(wt)
```

Here we construct a confidence interval for the mean which assumes $\sigma = 5$ is known.
```{r}

n <- length(wt)
xbar <- mean(wt)
alpha <- 0.05
za2 <- qnorm(1-alpha/2)
sigma <- 5


lo <- xbar - za2 * sigma / sqrt(n)
up <- xbar + za2 * sigma / sqrt(n)
```

The sample mean is $\bar X_n =  `r xbar`$.

The confidence interval is $[`r lo`,`r up`]$.


Now we construct a confidence interval without assuming $\sigma$ is known.
```{r}

sn <- sd(wt)
ta2 <- qt(1-alpha/2,n-1)
lot <- xbar - ta2 * sn/sqrt(n)
upt <- xbar + ta2 * sn/sqrt(n)
```

This confidence interval is $[`r lot`, `r upt`]$.


```{r}
mu0 <- 45
Ttest <- (xbar - mu0)/(sn/sqrt(n))
```


```{r}
mu0 <- 40
Ttest <- (xbar - mu0)/(sn/sqrt(n))
pt(Ttest,39-1)
```

A convenient way to get p-values and confidence intervals, etc. is the `t.test()` function:


```{r}
# two-sided
t.test(wt,mu=45)
# one-sided
t.test(wt,mu=40,alternative="less")
```

Draw power curves for a right-sided test.

```{r}
mu0 <- 45
alpha <- 0.05
sigma <- 5
n1 <- 10
mu <- seq(40,55,by = 0.01)
za <- qnorm(1 - alpha)
gm1 <- 1 - pnorm(za - (mu - mu0)/(sigma/sqrt(n1)))

n2 <- 20
gm2 <- 1 - pnorm(za - (mu - mu0)/(sigma/sqrt(n2)))


n3 <- 5
gm3 <- 1 - pnorm(za - (mu - mu0)/(sigma/sqrt(n3)))
plot(gm1~mu,type = "l")
lines(gm2~mu,lty=2)
lines(gm3~mu,lty=3)

abline(v=mu0,col="gray")
abline(h=alpha,col="gray")
```