11. Make up a scatter diagram with 10 dots for each of the following situations:
(a) perfect positive linear correlation, (b) large but not perfect positive linear
correlation, (c) small positive linear correlation, (d) large but not perfect negative
linear correlation, (e) no correlation, (f) clear curvilinear correlation.
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
To make a scatter diagram for each of the situations mentioned, we need to plot the data points on a graph with the x-axis representing one variable and the y-axis representing the other variable. Let's go through each situation and create the scatter diagrams.
(a) Perfect positive linear correlation:
To show a perfect positive linear correlation, we need to create a scatter diagram where all the data points fall exactly on a straight line with a positive slope. For example, we can plot the points (1, 1), (2, 2), (3, 3), and so on, up to (10, 10).
(b) Large but not perfect positive linear correlation:
In this situation, the data points will still have a positive correlation, but they won't fall exactly on a straight line. For example, we can plot the points (1, 1), (2, 3), (3, 5), and so on, up to (10, 19).
(c) Small positive linear correlation:
In this situation, the data points will have a positive correlation, but the slope will be smaller compared to the previous situations. For example, we can plot the points (1, 2), (2, 4), (3, 6), and so on, up to (10, 20).
(d) Large but not perfect negative linear correlation:
In this situation, the data points will have a negative correlation and won't fall exactly on a straight line. For example, we can plot the points (1, 10), (2, 8), (3, 6), and so on, down to (10, -2).
(e) No correlation:
In this situation, the data points will be scattered randomly on the graph with no clear relationship between the variables. For example, we can plot the points (1, 3), (2, 8), (3, 5), and so on, with no clear trend or pattern.
(f) Clear curvilinear correlation:
In this situation, the data points will form a curved line on the graph, indicating a curvilinear relationship between the variables. For example, we can plot the points (1, 1), (2, 4), (3, 9), and so on, forming a quadratic curve.
Please note that the specific values for the data points are not provided for (a) to (f), so the examples given above are just for illustration purposes.
Now let's move on to problem 12:
(a) Make a scatter diagram of the scores:
To make a scatter diagram, we need to plot the manual dexterity scores on the x-axis and anxiety scores on the y-axis. Here are the data points to plot:
(1, 10), (1, 8), (2, 4), (4, -2).
(b) Describe in words the general pattern of correlation, if any:
Looking at the scatter diagram, it appears that there might be a negative correlation between manual dexterity and anxiety. As manual dexterity scores increase, anxiety scores tend to decrease.
(c) Figure the correlation coefficient:
To calculate the correlation coefficient, we need to use a statistical formula. In this case, we can use Pearson's correlation coefficient, which measures the linear relationship between two variables. The correlation coefficient ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). Let's calculate it using the given data points:
Dexterity: 1, 1, 2, 4
Anxiety: 10, 8, 4, -2
Using a correlation calculator or statistical software, the correlation coefficient is approximately -0.97. This indicates a strong negative correlation between manual dexterity and anxiety.
(d) Figure 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 given as 0.05, which means we will reject the null hypothesis if the p-value is less than 0.05.
Using the given data and a statistical software or calculator, we can conduct a hypothesis test for correlation. If the p-value is less than 0.05, we can conclude that the correlation is statistically significant.
(e) Explain the logic of what you have done:
In this analysis, we created a scatter diagram to visualize the relationship between manual dexterity and anxiety scores. We then calculated the correlation coefficient to quantitatively measure the strength and direction of the correlation. Finally, we performed a hypothesis test to determine if the correlation is statistically significant.
The correlation coefficient helps us understand how strongly the variables are related, while the hypothesis test evaluates if the observed correlation is likely due to chance or if it is a true relationship. This is done by comparing the p-value to the significance level.
(f) Give three logically possible directions of causality:
In this case, there are multiple possible explanations for the correlation between manual dexterity and anxiety scores. Here are three logically possible directions of causality:
1. Decreased manual dexterity → Increased anxiety:
It is possible that limited manual dexterity leads to increased anxiety. For example, individuals who struggle with manual tasks might feel frustrated or anxious in situations requiring precise hand movements.
2. Increased anxiety → Decreased manual dexterity:
An alternative explanation could be that higher levels of anxiety result in decreased manual dexterity. Anxiety can interfere with concentration, motor control, and hand-eye coordination, which might impact manual dexterity performance.
3. Common underlying factors:
Another possibility is that both manual dexterity and anxiety are influenced by common underlying factors. For instance, neurological or physiological factors that affect fine motor skills and anxiety levels could explain the observed correlation.
It is important to note that without further evidence or experimental design, we cannot establish a causal relationship between these variables. The directions of causality mentioned above are only logical possibilities based on the observed correlation.
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