Some of the physiological effects of alcohol are well known. A paper by Squires et al. [1978] assessed the acute effects of alcohol on auditory brainstem potentials in humans. Six volunteers (including the three authors) participated in the study. The latency (delay) in response to an auditory stimulus was measured before and after an intoxicating dose of alcohol. Seven different peak responses were identified. In this exercise, we discuss only latency peak 3. Measurements of the latency of peak (in milliseconds after the stimulus onset) in the six subjects were as follows: Latency of Peak 1 2 3 4 5 6 Before alcohol 3.85 3.81 3.60 3.68 3.78 3.83 After alcohol 3.82 3.95 3.80 3.87 3.88 3.94 (a) Test the significance of the difference at the 0.05 level. (b) Calculate the p-value associated with the result observed. (c) Is your p-value based on a one- or two-tailed test? Why? (d) Carry out an (incorrect) two-sample test and state your conclusions. (e) Using the sample variances s and s associated with the set of readings observed before and after, calculate the variance of the difference, assuming independence (call this variance 1). How does this value compare with the variance of the difference calculated in part (a)? (Call this variance 2.) Why do you suppose variance 1 is so much bigger than variance 2? The average of the differences is the same as the difference in the averages. Show this. Hence, the two-sample t-test differed from the paired t-test only in the divisor. Which of the two tests in more powerful in this case, that is, declares a difference significant when in fact there is one?
Some of the physiological effects of alcohol are well known. A paper by Squires et al. [1978] assessed the acute effects of alcohol on auditory brainstem potentials in humans. Six volunteers (including the three authors) participated in the study. The latency (delay) in response to an auditory stimulus was measured before and after an intoxicating dose of alcohol. Seven different peak responses were identified. In this exercise, we discuss only latency peak 3. Measurements of the latency of peak (in milliseconds after the stimulus onset) in the six subjects were as follows: Latency of Peak 1 2 3 4 5 6 Before alcohol 3.85 3.81 3.60 3.68 3.78 3.83 After alcohol 3.82 3.95 3.80 3.87 3.88 3.94 (a) Test the significance of the difference at the 0.05 level. (b) Calculate the p-value associated with the result observed. (c) Is your p-value based on a one- or two-tailed test? Why? (d) Carry out an (incorrect) two-sample test and state your conclusions. (e) Using the sample variances s and s associated with the set of readings observed before and after, calculate the variance of the difference, assuming independence (call this variance 1). How does this value compare with the variance of the difference calculated in part (a)? (Call this variance 2.) Why do you suppose variance 1 is so much bigger than variance 2? The average of the differences is the same as the difference in the averages. Show this. Hence, the two-sample t-test differed from the paired t-test only in the divisor. Which of the two tests in more powerful in this case, that is, declares a difference significant when in fact there is one?
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![Some of the physiological effects of alcohol are well known. A paper by Squires et al.
[1978] assessed the acute effects of alcohol on auditory brainstem potentials in humans.
Six volunteers (including the three authors) participated in the study. The latency (delay)
in response to an auditory stimulus was measured before and after an intoxicating dose of
alcohol. Seven different peak responses were identified. In this exercise, we discuss only
latency peak 3. Measurements of the latency of peak (in milliseconds after the stimulus
onset) in the six subjects were as follows:
Latency of Peak
1
2
3
4
5
6.
Before alcohol
3.85
3.81
3.60
3.68
3.78
3.83
After alcohol
3.82
3.95
3.80
3.87
3.88
3.94
(a) Test the significance of the difference at the 0.05 level.
(b) Calculate the p-value associated with the result observed.
(c) Is your p-value based on a one- or two-tailed test? Why?
(d) Carry out an (incorrect) two-sample test and state your conclusions.
(e) Using the sample variances s? and s associated with the set of readings observed
before and after, calculate the variance of the difference, assuming independence
(call this variance 1). How does this value compare with the variance of the
difference calculated in part (a)? (Call this variance 2.) Why do you suppose
variance 1 is so much bigger than variance 2? The average of the differences
is the same as the difference in the averages. Show this. Hence, the two-sample
t-test differed from the paired t-test only in the divisor. Which of the two tests in
more powerful in this case, that is, declares a difference significant when in fact
there is one?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ff3b9ae-2a61-43b1-a234-42de4cceedd2%2Ffbba56f4-e5ea-4922-a60e-22ecca44f904%2Fu2obuc_processed.png&w=3840&q=75)
Transcribed Image Text:Some of the physiological effects of alcohol are well known. A paper by Squires et al.
[1978] assessed the acute effects of alcohol on auditory brainstem potentials in humans.
Six volunteers (including the three authors) participated in the study. The latency (delay)
in response to an auditory stimulus was measured before and after an intoxicating dose of
alcohol. Seven different peak responses were identified. In this exercise, we discuss only
latency peak 3. Measurements of the latency of peak (in milliseconds after the stimulus
onset) in the six subjects were as follows:
Latency of Peak
1
2
3
4
5
6.
Before alcohol
3.85
3.81
3.60
3.68
3.78
3.83
After alcohol
3.82
3.95
3.80
3.87
3.88
3.94
(a) Test the significance of the difference at the 0.05 level.
(b) Calculate the p-value associated with the result observed.
(c) Is your p-value based on a one- or two-tailed test? Why?
(d) Carry out an (incorrect) two-sample test and state your conclusions.
(e) Using the sample variances s? and s associated with the set of readings observed
before and after, calculate the variance of the difference, assuming independence
(call this variance 1). How does this value compare with the variance of the
difference calculated in part (a)? (Call this variance 2.) Why do you suppose
variance 1 is so much bigger than variance 2? The average of the differences
is the same as the difference in the averages. Show this. Hence, the two-sample
t-test differed from the paired t-test only in the divisor. Which of the two tests in
more powerful in this case, that is, declares a difference significant when in fact
there is one?
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