01-02-Stat-Methods-Hypothesis-Testing.Rmd

## Hypothesis Testing {-}
  * Shapiro Wilks
  * One Sample T-Test
  * Calculating Power
  * Hypothesis Testing
  * Sample Size Determination


```{r a5, comment=NA, fig.width=8, fig.height=5}

## Data sample on chick weights
data(chickwts)
head(chickwts)

## Normal reference plot for height
qqnorm(chickwts$weight)
qqline(chickwts$weight)

## Are the data from a normal distribution?
shapiro.test(chickwts$weight)
mean(chickwts$weight)

```

How does the sample mean compare to a hypothesis test that the true mean is < 260? What is the power of the test?

$$H_0: \mu \ge 260, H_a: \mu \lt 260$$

|  Population   | Fail to Reject | Reject $H_0$ |
|:-------------:|:--------------:|:------------:|
| $H_0$ is True |    Correct     | Type I Error |
| $H_a$ is True | Type II Error  |   Correct    |

```{r a6, comment=NA, warning=FALSE, message=FALSE}

## What is the probability of a Type I error if we say the true mean is less than 250?
t.test(chickwts$weight, mu = 250, alternative = "less")

## Verify the t statistic and p-value
(ts = (mean(chickwts$weight) - 250) / (sd(chickwts$weight) / sqrt(length(chickwts$weight))))
pt(ts, df = 70)

## What is the probability of a Type I error if we say the true mean is > 245?
t.test(chickwts$weight, mu = 245, alternative = "greater")

## Verify the t statistic and p-value
(ts = (mean(chickwts$weight) - 245) / (sd(chickwts$weight) / sqrt(length(chickwts$weight))))
1 - pt(ts, df = 70)

## We have rejected the null hypothesis and said under an alpha of .05 there is enough evidence
## to suppor that the true mean of Chick Weights is > 245

## What is the power of our test?
power.t.test(n = length(chickwts$weight),
             delta = abs(mean(chickwts$weight) - 245),
             sd = sd(chickwts$weight),
             sig.level = .05,
             type = "one.sample",
             alternative = "one.sided", strict = TRUE)

## What sample size would we need to have a power of .8?
power.t.test(delta = abs(mean(chickwts$weight) - 245),
             sd = sd(chickwts$weight),
             sig.level = .05,
             power = .8,
             type = "one.sample",
             alternative = "one.sided", strict = TRUE)

## Verify manually
(sd(chickwts$weight)^2 * (qnorm(p = .95) + qnorm(p = .8))^2) / abs(mean(chickwts$weight) - 245)^2
```