For problems 12 to 14, do the following: (a) Make a scatter diagram of the
scores; (b) describe in words the general pattern of correlation, if any; (c) figure
the correlation coefficient; (d) figure whether the correlation is statistically significant
(use the .05 significance level, two-tailed); (e) explain the logic of what
you have done, writing as if you are speaking to someone who has never heard
of correlation (but who does understand the mean, deviation scores, and hypothesis
testing); and (f) give three logically possible directions of causality, indicating
for each direction whether it is a reasonable explanation for the correlation
in light of the variables involved (and why).
12. Four research participants take a test of manual dexterity (high scores mean better dexterity)
and an anxiety test (high scores mean more anxiety). The scores are as follows.
(c)
(e)
(d)
(f)
Person Dexterity Anxiety
1 1 10
2 1 8
3 2 4
4 4 -2
13. Four
a) To make a scatter diagram for the scores, you would need to plot the points on a graph. The x-axis represents the scores for manual dexterity, and the y-axis represents the scores for anxiety.
b) The general pattern of correlation, if any, can be described by looking at the scatter diagram. If there is a positive correlation, it means that as the scores for manual dexterity increase, the scores for anxiety also tend to increase. If there is a negative correlation, it means that as the scores for manual dexterity increase, the scores for anxiety tend to decrease. If there is no clear pattern, it means that there is no correlation between the two variables.
c) To figure the correlation coefficient, you would need to calculate the covariance and the standard deviations for both variables. You can then use these values to calculate the correlation coefficient using the formula: correlation coefficient = covariance / (standard deviation of variable 1 * standard deviation of variable 2).
d) To figure out whether the correlation is statistically significant, you would need to perform a hypothesis test. You would set up the null hypothesis that there is no correlation between the two variables and the alternative hypothesis that there is a correlation. You would then use a significance level of 0.05 (or any other predetermined level) to determine whether to reject the null hypothesis in favor of the alternative hypothesis.
e) The logic of correlation is that it measures the strength and direction of the relationship between two variables. It allows us to determine if there is a systematic relationship between the two variables and to quantify this relationship. By calculating the correlation coefficient and performing a hypothesis test, we can determine if the relationship we observe in the data is statistically significant or if it could have occurred by chance.
f) Three logically possible directions of causality for the correlation could be:
- Manual dexterity causing anxiety: It is possible that individuals with better manual dexterity may be more confident and less anxious during tasks that require manual skills.
- Anxiety causing a decrease in manual dexterity: It is possible that individuals with higher levels of anxiety may struggle more with tasks that require manual dexterity due to increased stress or nervousness.
- Common underlying factors: It is also possible that there may be other factors that influence both dexterity and anxiety, leading to a correlation between the two variables. For example, a neurological condition may affect both manual dexterity abilities and anxiety levels.
To answer the question, we need to follow the steps outlined in the instructions:
(a) Make a scatter diagram of the scores:
A scatter diagram is a graphical representation that shows the relationship between two variables. In this case, we have two variables: manual dexterity and anxiety. Plot the scores of manual dexterity on the x-axis and anxiety on the y-axis. For example, the first data point (1, 10) represents a manual dexterity score of 1 and an anxiety score of 10. Repeat this for all data points.
(b) Describe in words the general pattern of correlation, if any:
Look at the scatter diagram and observe the overall pattern. Does it look like there is a linear relationship between the two variables? If the points on the scatter diagram tend to form a line or a curve, we can say there is a correlation. If they do not appear to have any specific pattern, we can say there is no correlation.
(c) Calculate the correlation coefficient:
The correlation coefficient measures the strength and direction of the linear relationship between two variables. To calculate it, we can use a statistical software or calculator. If you have the data points, you can use a formula. The formula for the correlation coefficient is a bit complex, so it is best to use a calculator or software.
(d) Determine whether the correlation is statistically significant:
To determine if the correlation is statistically significant, we need to perform a hypothesis test. The significance level is set at 0.05, meaning we want a 5% chance of a Type I error. We can use a statistical test like a t-test or a correlation test to analyze the data and determine the significance.
(e) Explain the logic of what you have done, assuming the person has no knowledge of correlation:
Correlation measures the relationship between two variables. By making a scatter diagram, we visually examine the data points to see if there is a relationship. The correlation coefficient gives us a numerical measure of the strength and direction of this relationship. We then perform a statistical test to determine if the observed relationship is significant, meaning it is unlikely to occur by chance.
(f) Give three logically possible directions of causality:
Causality refers to cause and effect relationships between variables. Here are three possible directions of causality for the correlation between manual dexterity and anxiety:
1. Manual dexterity causing anxiety: It is possible that individuals with poor manual dexterity experience more anxiety due to frustration or limitations in performing tasks.
2. Anxiety causing poor manual dexterity: It is possible that individuals with high levels of anxiety struggle with fine motor skills, leading to lower manual dexterity scores.
3. Common underlying factor: It is possible that there is a third factor influencing both manual dexterity and anxiety. For example, high levels of stress or a neurological condition could impact both variables.