# Exercise 1

Suppose that a researcher is interested in the effect of caffeine on typing speed. A group of nine individuals are administered a typing test. The following day, they repeat the typing test, this time after taking 400 mg of caffeine. (Note: This is not recommended.) The data gathered, measured in words per minute, is

decaf = c(98,  124, 107, 105, 80, 43, 73, 68, 69)
caff  = c(104, 128, 110, 108, 86, 53, 72, 73, 72)
##   decaf caff
## 1    98  104
## 2   124  128
## 3   107  110
## 4   105  108
## 5    80   86
## 6    43   53
## 7    73   72
## 8    68   73
## 9    69   72

Note that these are paired observations.

Use the sign test with a significance level of 0.05 to assess whether or not caffeine has an effect on typing speed. That is, test

$H_0\colon \ m_D = m_C - m_N = 0 \quad \text{vs} \quad H_A\colon \ m_D = m_C - m_N \neq 0$

where

• $$m_C$$ is the median typing speed in words per minute of individuals using caffeine
• $$m_N$$ is the median typing speed in words per minute of individuals not using caffeine

Since it is possible that the caffeine makes typing speed worse, use a two-sided test. (Also note that this is a silly experience, we aren’t considering typing accuracy!)

Report:

• The value of the test statistic for the observed data.
• The distribution of the test statistic under the null hypothesis.
• The p-value of the test.
• A decision when $$\alpha = 0.05$$.
• A conclusion in words.

# Exercise 2

Does meditation have an effect on blood pressure. A group of six college aged individuals were given a routine physical examination including a measurement of their systolic blood pressure. (Measured in millimeters of mercury.) A week after their physicals, the same six individuals returned for a guided meditation session. Immediately afterwords there (systolic) blood pressure was measured. The data gathered is

physical    = c(125, 108, 185, 135, 112, 133)
meditation  = c(120, 114, 160, 131, 124, 125)
##   physical meditation
## 1      125        120
## 2      108        114
## 3      185        160
## 4      135        131
## 5      112        124
## 6      133        125

Note that these are paired observations.

Use the sign test with a significance level of 0.10 to assess whether or not meditation has an effect on blood pressure. That is, test

$H_0\colon \ m_D = m_M - m_P = 0 \quad \text{vs} \quad H_A\colon \ m_D = m_M - m_P \neq 0$

where

• $$m_P$$ is the median systolic blood pressure in millimeters of mercury measured without meditation
• $$m_M$$ is the median systolic blood pressure in millimeters of mercury measured with meditation

Since it is possible that the meditation makes blood pressure worse, use a two-sided test.

Report:

• The value of the test statistic for the observed data.
• The distribution of the test statistic under the null hypothesis.
• The p-value of the test.
• A decision when $$\alpha = 0.10$$.
• A conclusion in words.

# Exercise 3

• $$H_0$$: The distribution of systolic blood pressure is the same with and without meditation
• $$H_A$$: The distribution of systolic blood pressure is different with and without meditation

To do so, use a permutation test that permutes the statistic

$\bar{x}_D$

where $$\bar{x}_D$$ is the sample mean difference. Assume that the distribution of blood pressure with and without meditation has the same shape, but may have different locations. Use at least 10000 permutations.

physical    = c(125, 108, 185, 135, 112, 133)
meditation  = c(120, 114, 160, 131, 124, 125)
• Create a histogram that illustrates the distribution of the statistic used.
• Report the p-value of the test.

# Example 4

Which profession pays more? Data Scientist or Actuary? A (far too small) survey of junior (less than three years experience) data scientists and actuaries resulted in the following data:

data_sci = c(88000, 121000, 91000, 50000, 78000, 95000)
actuary = c(63000, 75000, 81000, 75000, 85000)

Use a permutation test that permutes the statistic

$t = \frac{(\bar{x} - \bar{y}) - 0}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

to test

• $$H_0$$: The distribution of salaries is the same for junior data scientists and actuaries
• $$H_A$$: The distribution of salaries is different for junior data scientists and actuaries

Assume that the distribution of salaries for both has the same shape, but may have different locations. Use at least 10000 permutations.

• Create a histogram that illustrates the distribution of the statistic used.
• Report the p-value of the test.

# Exercise 5

Repeat exercise 3, but use an appropriate test available in the R function wilcox.test().

Report:

• The p-value of the test
• A decision when $$\alpha = 0.05$$.