A scientist wishes to investigate whether exposure to sunlight reduces the amount of time it takes for a particular reaction to take place. There is natural variability in reaction time, and data are recorded for 10 instances of reaction time in a dark environment, and 10 instances in sunlight. These are presented in the Table . Experiment Conditions Time Experiment Conditions Time 1 Bright 7.1 11 Dark 7.4 2 Bright 6.2 12 Dark 7 3 Bright 8.1 13 Dark 8.1 4 Bright 7.4 14 Dark 8.9 5 Bright 7.2 15 Dark 7.1 6 Bright 6.4 16 Dark 7 7 Bright 6.5 17 Dark 7.5 8 Bright 6.7 18 Dark 8.6 9 Bright 6.8 19 Dark 7.3 10 Bright 8 20 Dark 6.9 Table . Reaction times The scientist does a statistical test and gets output as given in the Figure . The scientist really believes that the data confirm the hypothesis that reactions in sunlight are on average quicker, but feels that since the statistical test showed no significant result it will not be possible to publish anything. The scientist then goes to see a statistician to find out if there is anything that ‘can be done’ to make the results significant. The statistician flippantly suggests a one sided test would ‘do the trick’, but emphasizes that a better solution by far would involve collecting more data. State the true research question of interest, and the corresponding null and alternate hypotheses. What is the null hypothesis being tested by the analysis the scientist has done and how does it differ from the one you have stated? Explain the two types of error that may be made when testing, and how the sample size affects the probability of making each one, if a predetermined ‘significance level’ of 5% is chosen? The scientist would like to estimate the difference between the average reaction times to within 0.3 seconds. (95% confidence interval width 0.6 s). It is agreed that more experiments will be run. Do a sample size calculation to determine how many more experiments of each type should be run
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
A scientist wishes to investigate whether exposure to sunlight reduces the amount of time it takes for a particular reaction to take place. There is natural variability in reaction time, and data are recorded for 10 instances of reaction time in a dark environment, and 10 instances in sunlight.
These are presented in the Table .
|
Experiment |
Conditions |
Time |
|
Experiment |
Conditions |
Time |
|
1 |
Bright |
7.1 |
11 |
Dark |
7.4 |
|
|
2 |
Bright |
6.2 |
12 |
Dark |
7 |
|
|
3 |
Bright |
8.1 |
13 |
Dark |
8.1 |
|
|
4 |
Bright |
7.4 |
14 |
Dark |
8.9 |
|
|
5 |
Bright |
7.2 |
15 |
Dark |
7.1 |
|
|
6 |
Bright |
6.4 |
16 |
Dark |
7 |
|
|
7 |
Bright |
6.5 |
17 |
Dark |
7.5 |
|
|
8 |
Bright |
6.7 |
18 |
Dark |
8.6 |
|
|
9 |
Bright |
6.8 |
19 |
Dark |
7.3 |
|
|
|
|
|
|
|
|
10 Bright 8 20 Dark 6.9
Table . Reaction times
The scientist does a statistical test and gets output as given in the Figure . The scientist really believes that the data confirm the hypothesis that reactions in sunlight are on average quicker, but feels that since the statistical test showed no significant result it will not be possible to publish anything. The scientist then goes to see a statistician to find out if there is anything that ‘can be done’ to make the results significant. The statistician flippantly suggests a one sided test would ‘do the trick’, but emphasizes that a better solution by far would involve collecting more data.
- State the true research question of interest, and the corresponding null and alternate hypotheses. What is the null hypothesis being tested by the analysis the scientist has done and how does it differ from the one you have stated?
- Explain the two types of error that may be made when testing, and how the sample size affects the probability of making each one, if a predetermined ‘significance level’ of 5% is chosen?
- The scientist would like to estimate the difference between the average reaction times to within 0.3 seconds. (95% confidence interval width 0.6 s). It is agreed that more experiments will be run. Do a sample size calculation to determine how many more experiments of each type should be run.
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