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)

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.

You may use the following probabilities calculated in R.

round(dbinom(x = 0:9, size = 9, prob = 0.5), 3)

##  [1] 0.002 0.018 0.070 0.164 0.246 0.246 0.164 0.070 0.018 0.002

Report:

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

# use this chunk to complete any necessary calculations in R

• P-Value: Your p-value here.
• Decision: Your decision here.

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)

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.

You may use the following probabilities calculated in R.

round(dbinom(x = 0:6, size = 6, prob = 0.5), 3)

## [1] 0.016 0.094 0.234 0.312 0.234 0.094 0.016

Report:

• The p-value of the test
• A decision when \(\alpha = 0.10\).

# use this chunk to complete any necessary calculations in R

• P-Value: Your p-value here.
• Decision: Your decision here.

Exercise 3

Return to the sleep data in Exercise 2. This time test

• \(H_0\): 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.

# use this chunk to complete any necessary permutation calculations
# also calculate statistic on observed data

# use this chunk to create the histogram

# use this chunk to calculate the p-value of the test

Example 4

Which profession pays more? Data Scientist of Actuary? A (far too small) survey of junior (less than three years experience) data scientist 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.

# use this chunk to complete any necessary permutation calculations
# also calculate statistic on observed data

# use this chunk to create the histogram

# use this chunk to calculate 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 \(\alpha = 0.05\).

# use this chunk to complete any necessary calculations in R

• P-Value: Your p-value here.
• Decision: Your decision here.