Lab11Worksheet.Rmd

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title: "DATA1001 Lab11 Worksheet"
subtitle: "Tests for a Mean and Relationships"
author: "© University of Sydney 2020"
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# <i class="fa fa-crosshairs"></i> Aim {-}

At the end of this Worksheet, you should be able to:

- use the "Box Model" and the Hypothesis Testing framework to perform a 1 sample Z or T Test.
- use R Output from `t.test` to perform a 1 or 2 sample T Test.

<br>

**Recap on the 1 and 2 sample T Tests**

- The structure of a hypothesis test is H A T P C.

- See [Lecture](http://www.maths.usyd.edu.au/u/UG/JM/DATA1001/r/current/lectures/33.ZandTtests.html) and [Lecture](http://www.maths.usyd.edu.au/u/UG/JM/DATA1001/r/current/lectures/34.2SampleTTest.html).

- For a box model contructed around a certain null hypothesis, the **observed value of the test statistic** is **(OV - EV)/SE**, where the OV is calculated from the sample and the EV and SE are calculated from the box model.

-  For a 1 Sample Z or T Test, the null hypothesis is about the population **mean**, so we consider the Sample **Mean**, and the test statistic has a Z or T distribution respectively, depending on whether we know the population SD.

Test|Test Statistic|P-value Curve|
|---|------------------------|------|
|Z| $\frac{\mbox{observed mean - population mean}}{\mbox{population SD}/\sqrt{n}}$|Normal|
|T| $\frac{\mbox{observed mean - population mean}}{\mbox{sample SD}/\sqrt{n}}$|$t_{n-1}$|

- Most commonly, we use the T Test. The easiest way is to use `t.test()` in R, which calculates the test statistics and p-values.

<br>
**When you finish:** Upload your answers to the Lab12 link on [Ed](https://edstem.org/courses/3894/discussion/).

<br><br>

# Have a go

Explain the structure of a T Test using an annotated diagram, and example.

<br><br><br><br><br><br>


<br><br>

# `r icon::fa("bicycle")` Caffeine and endurance (1 Sample Z Test)

A [study](https://www.ncbi.nlm.nih.gov/pubmed/7657415) considered caffeine effect's on endurance, with a double blind, random order administration of caffeine capsules for 9 elite cylists.  

The following data is the time to exhaustion after  0 and 13 mg caffeine per kg body weight, with `cafdiff` representing the effect of caffeine on endurance. 

Assume that we know the population SD is 12 mins, from a previous study.

```{r}
caf0 = c(36.05, 52.47, 56.55, 45.2, 35.25, 66.38, 40.57, 57.15, 28.34)  # no caffeine (base-line endurance)
caf13 = c(37.55, 59.3, 79.12, 58.33, 70.54, 69.47, 46.48, 66.35, 36.2)  # 13 mg caffeine per kg body weight
cafdiff = caf13- caf0  # This represents the 'effect' of caffeine on endurance.
cafdiff
```

Test the claim that caffeine does *not* affect endurance - ie the mean `cafdiff` is 0.

Note:

- We define "endurance" to be  "time to exhausation" here.
- This is a 1 sample Z test, as we are using the 12 values in `cafdiff`, and we know the population SD = 12.

## Preparation

Draw a simple box model, with any population and sample details identified.

<br><br><br><br><br><br><br><br><br><br>

## Hypothesis Test


### H {-}

- Write down the null and alternative hypotheses.

Ho:  The mean effect of caffeine on endurance (time to exhausation) is 0.

H1:  The mean effect of caffeine on endurance is not 0.

- Add the null hypothesis to the box model above, with the specific mean of the box. As we have the population SD, we can use the Z Test. 

### A {-}

Write down any assumptions.

### T {-}

- For a sample of size 12 from the box, what is the expected value (EV) and standard error (SE) of the **Mean** of the sample?

```{r}
n = 9
ev = 0    # from Ho in box model
se = 12/sqrt(n)
```

<br><br>

- What is the formula for the test statistic?

- What is the observed value of the test statistic?

```{r}
ov = mean(cafdiff)
ts = (ov-ev)/se
ts
```

### P {-}

Using the Normal curve curve to model the **Mean** of the sample, what is the approximate p-value?

```{r}
2*pnorm(ts,lower.tail=F)
```

<br><br>

### C {-}

What is the conclusion?  

<br><br>

# `r icon::fa("bicycle")` Caffeine and endurance (1 Sample T Test)

A [study](https://www.ncbi.nlm.nih.gov/pubmed/7657415) considered caffeine effect's on endurance, with a double blind, random order administration of caffeine capsules for 9 elite cylists.  

The following data is the time to exhaustion after  0 and 13 mg caffeine per kg body weight, with `cafdiff` representing the effect of caffeine on endurance. 


```{r}
caf0 = c(36.05, 52.47, 56.55, 45.2, 35.25, 66.38, 40.57, 57.15, 28.34)  # no caffeine (base-line endurance)
caf13 = c(37.55, 59.3, 79.12, 58.33, 70.54, 69.47, 46.48, 66.35, 36.2)  # 13 mg caffeine per kg body weight
cafdiff = caf13- caf0  # This represents the 'effect' of caffeine on endurance.
cafdiff
```

Test the claim that caffeine does *not* affect endurance - ie the mean `cafdiff` is 0.

Note: Now, we do not know the population SD, so we will perform a 1 sample T Test.


## Preparation

Draw a simple box model, with any population and sample details identified.

<br><br><br><br><br><br><br><br><br><br>

## Hypothesis Test


### H {-}

- Write down the null and alternative hypotheses.

Ho:  The mean effect of caffeine on endurance (time to exhausation) is 0.

H1:  The mean effect of caffeine on endurance is not 0.

- Add the null hypothesis to the box model above, with the specific mean of the box. As we do not have the population SD, we will use the T Test.


### A {-}

Write down any assumptions.

### T {-}

- For a sample of size 12 from the box, what is the expected value (EV) and standard error (SE) of the **Mean** of the sample?

```{r}
n = 9
ev = 0    # from Ho in box model
se = sd(cafdiff)/sqrt(n)
```

<br><br>

- What is the formula for the test statistic?

- What is the observed value of the test statistic?

```{r}
ov = mean(cafdiff)
ts = (ov-ev)/se
ts
```

### P {-}

Using a $t_{11}$ curve curve to model the **Mean** of the sample, what is the approximate p-value?

```{r}
2*pt(ts,n-1, lower.tail=F)
```


<br><br>

### C {-}

What is the conclusion?  

<br><br>

The speedy way to do all this is:
```{r}
t.test(mu = 0, cafdiff)
```

<br><br>





<br><br>

# `r icon::fa("heartbeat")` Caffeine and endurance (2 Sample T Test)

Consider the following data on heart rates (beats per minute), for 2 independent groups of Sydney students, collected 20 minutes after the 'RedBull' group had drunk a 250ml cold can of Red Bull.

No_RB = 84,76,68,80,64,62,74,84,68,96,80,64,65,66

RB = 72,88,72,88,76,75,84,80,60,96,80,84

```{r, echo=F, warning=F, message=F}
# Unequal length 2-sample T-Test

No_RB = c(84,76,68,80,64,62,74,84,68,96,80,64,65,66)
RB = c(72,88,72,88,76,75,84,80,60,96,80,84)

# Create data frame
RB_data <- data.frame(group = c(rep("No_RB",14), rep("RB",12)),rate = c(No_RB,  RB))
```

Test the claim that Redbull (caffeine) has an effect on heart rate

Note: 
- We are comparing 2 independent populations, so we will use a 2 sample T Test.
- You can use the information given in `t.test`.

```{r}
No_RB = c(84,76,68,80,64,62,74,84,68,96,80,64,65,66)
RB = c(72,88,72,88,76,75,84,80,60,96,80,84)
t.test(No_RB, RB, var.equal = T)
```

## Preparation

Draw a simple box model with the 2 populations and 2 samples, with any population and sample details identified.

<br><br><br><br><br><br><br><br><br><br>

## Hypothesis Test


### H {-}

- Write down the null and alternative hypotheses.

Ho:  The mean difference between the heart rates of the 2 populations (with and without Red Bull) is 0.

H1:  The mean difference between the heart rates of the 2 populations (with and without Red Bull) is not 0.

- Add the null hypothesis to the box model above, with the specific mean difference of the boxes.

### A {-}

Write down any assumptions.

### T {-}

From the R Output, what is the observed value of the test statistic?

### P {-}

From the R Output, what is the p-value?

<br><br>

### C {-}

What is the conclusion?  

<br><br>

## Diagnostic checks for the assumptions

**A1. The 2 samples are independent**

Given in context.

<br>


**A2. The 2 populations have equal spread (SD/variance)**

```{r, echo = F, fig.height= 3, warning=FALSE}
require(ggplot2)

p2 <- ggplot(RB_data, aes(x = group, y = rate)) +
  geom_boxplot(fill = c(2,3), colour = 1) +
  xlab("Group") +
  ylab("Heart rate (bpm)") +
  ggtitle("Boxplot of effect of Red Bull on student heart rate")
p2
```

```{r}
var.test(No_RB,RB)
```

<br>

**A3. The 2 populations are Normal**

Boxplots look fairly symmetric.

```{r, echo = F, fig.height= 3, warning=FALSE}
require(ggplot2)
p3 = ggplot(RB_data, aes(sample = rate, colour = group)) +  
  stat_qq() + stat_qq_line() + ggtitle("QQplot")
p3
```

```{r,eval=F}
require(ggplot2)
p3 = ggplot(RB_data, aes(sample = rate, colour = group)) +  
  stat_qq() + stat_qq_line() + ggtitle("QQplot")
p3
```

```{r}
shapiro.test(No_RB)
shapiro.test(RB)
```