In jr-packages/jrAnalytics: Practicals and datasets for Jumping Rivers courses

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Simple Hypothesis Testing

Method A

---

78.64 79.01 79.57 79.52 80.71 79.95 78.50 79.10 81.98 80.09 80.29 80.22

---

Method B

---

81.92 81.12 82.47 82.86 82.89 82.45 82.51 81.11 83.07 82.77 82.38 83.14

---

1. We conducted an experiment and collected the data in the tables above. This data set isn't paired.^[I intentionally didn't make the data available for download so you would have to think about how to enter the data. You could enter it either Excel and import or directly into R.] a) Input the data into ^[Here I would suggest input the data into Excel and using read_csv()]. Combine the two data sets into a single data frame.

   ```r

   Data for question 1

   Easier using Excel and export as CSV

   x = c(78.64, 79.01, 79.57, 79.52, 80.71, 79.95, 78.50, 79.10, 81.98, 80.09, 80.29, 80.22) y = c(81.92, 81.12, 82.47, 82.86, 82.89, 82.45, 82.51, 81.11, 83.07, 82.77, 82.38, 83.14) dd = data.frame(x, y)

   ```

   r d1 = data.frame(value = x) d2 = data.frame(value = y)

   ```r

   Suppose you have two separate data files. Here is some code that

   will help you combine them. First we read in the separate files:

   d1 = read.csv("Method1.csv") d2 = read.csv("Method2.csv") ```

   ```r

   In order to combine the data frames,

   they must have the same column names:

   head(d1, 2) head(d2, 2) ```

   ```r

   We combine data frames using rbind (row bind)

   d = rbind(d1, d2) ```

   ```r

   Finally we create a new column to indicate the Method

   rep is the replicate function. See ?rep

   d$Method = rep(1:2, each = 12) head(d, 2) ```

   b) Exploratory data analysis. Construct boxplots, histograms and q-q plots for both data sets. Work out the means and standard deviations. Before carrying out any statistical test, what do you think your conclusions will be? Do you think the variances are roughly equal? Do you think the data conforms to a normal distribution. c) Carry out a two sample $t$-test. Assume that the variances are unequal.

   r t.test(value ~ Method, data = d, var.equal = FALSE)

   How does this answer compare with your intuition?

   d) Carry out a two sample $t$-test, assuming equal variances.

   r t.test(value ~ Method, data = d, var.equal = TRUE)

2. Suppose we are interested whether successful business executives are affected by their zodiac sign. We have collected 4265 samples and obtained the following data

   r data(zodiac, package = "jrAnalytics") head(zodiac)

   a) Carry out a $\chi^2$ goodness of fit test on the zodiac data. Are business executives distributed uniformly across zodiac signs?

   ```r x = zodiac$count m = chisq.test(x)

   Since p > 0.05 we can't accept the alternative hypothesis.

   However, the question is worded as though we can "prove" the Null

   hypotheis, which we obviously can't do.

   ```

   b) What are the expected values for each zodiac sign?

   ```r

   expected values

   (expected = m[["expected"]]) ```

   c) The formula for calculating the residuals ^[These residuals are called Pearson residuals. Hint: use str(m) to extract the residuals.] is given by

   $$ \frac{\text{observed} - \text{expected}}{\sqrt{\text{expected}}} $$

   Which residuals are large?

   ```r

   Residuals

   m[["residuals"]] ```

One way ANOVA tables

1. A pilot study was developed to investigate whether music influenced exam scores. Three groups of students listened to 10 minutes of Mozart, silence or heavy metal before an IQ test. The results of the IQ test are as follows

---

Mozart 109 114 108 123 115 108 114 Silence 113 114 113 108 119 112 110 Heavy Metal 103 94 114 107 107 113 107

---

a) Construct a one-way ANOVA table. Are there differences between treatment groups?

```r
x1 = c(109, 114, 108, 123, 115, 108, 114)
x2 = c(113, 114, 113, 108, 119, 112, 110)
x3 = c(103, 94, 114, 107, 107, 113, 107)
dd = data.frame(values = c(x1, x2, x3),
                type = rep(c("M", "S", "H"),
                each = 7))
m = aov(values ~ type, dd)
summary(m)
##The p value is around 0.056.
##This suggests a difference may exist.
```

b) Check the standardised residuals of your model.

```r
plot(fitted.values(m), rstandard(m))
## Residual plot looks OK

```

c) Perform a multiple comparison test to determine where the difference lies.

```r
TukeyHSD(m)
```

jr-packages/jrAnalytics documentation built on Jan. 27, 2020, 5:04 a.m.