2.7 Exercises

2.7.1 Practice using qt

Take an independent random sample of size 142 from a normal distribution with mean 123, and standard deviation 70. Next, we are going to pretend we don’t know the population parameters (the mean and standard deviation). We compute the MLEs of the mean and standard deviation using the data and get the sample mean 145.242 and the sample standard deviation 50.885.

• Compute the estimated standard error using the sample standard deviation provided above.
• What are your degrees of freedom for the relevant t-distribution?
• Calculate the absolute critical t-value for a 95% confidence interval using the relevant degrees of freedom you just wrote above.
• Next, compute the lower bound of the 95% confidence interval using the estimated standard error and the critical t-value.
• Finally, compute the upper bound of the 95% confidence interval using the estimated standard error and the critical t-value.

2.7.2 Computing the p-value

A paired t-test is done with data from 10 participants. The t-value from the test is 2.1. What is the p-value associated with a two-sided null hypothesis test?

2.7.3 Computing the t-value

If the p-value from a two-sided null hypothesis test had been 0.09, what would be the associated absolute t-value (i.e., ignoring the sign on the t-value)? The number of participants is 10, as above.

2.7.4 Type I and II error

Given that Type I error is 0.01; what is the highest value possible for Type II error?

2.7.5 Practice with the paired t-test

In a self-paced reading study, Grodner and Gibson (2005) investigated subjects vs. object relative clauses. They analyzed the reading times at the relative clause verb. However, a reviewer objects that the whole sentence’s reading times (total reading times) should be used to evaluate the difference between the two conditions, because one cannot know where the difficulty might arise. It isn’t clear whether one should use mean reading times over the entire sentence, or total reading times (summing up all the reading times over the entire sentence). Carry out a by-subjects paired t-test on (a) the critical relative clause verb, versus (b) mean reading time over all words in the two sentence types, and (c) total reading times over all words in the two sentence types. Compare the t-value across the three tests, and decide what the appropriate dependent variable might be (Note: there is no correct answer here).

The data are loaded and pre-processed as follows. The code below gives you the reading times for the critical verb. You will have to work out how to obtain mean or total reading times for the whole sentence in each condition.

## load data:
library(dplyr)
gg05e1 <- read.table("data/GrodnerGibson2005E1.csv",sep=",", 
                     header=TRUE)
gge1 <- gg05e1 %>% 
  filter(item != 0)

gge1 <- gge1 %>% mutate(word_positionnew = ifelse(item != 15 & 
                                                    word_position > 10,
                                                  word_position-1, word_position)) 
#there is a mistake in the coding of word position,
#all items but 15 have regions 10 and higher coded
#as words 11 and higher

## get data from relative clause verb:
gge1crit <- subset(gge1, ( condition == "objgap" & 
                             word_position == 6 ) |
            ( condition == "subjgap" & word_position == 4 ))

References

Grodner, Daniel, and Edward Gibson. 2005. “Consequences of the Serial Nature of Linguistic Input.” Cognitive Science 29: 261–90.