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.
Person Dexterity Anxiety
1 1 10
2 1 8
3 2 4
4 4 -2
(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).
Please I need help understanding this!!!
Part F is what i need help on this is due in a few hours please help!! Thanks
http://www.google.com/search?client=safari&rls=en&q=Calculating+correlation+coefficient&ie=UTF-8&oe=UTF-8
Although the sample is very small, it is likely to give a positive correlation. (f) If there is a correlation, it can be that A causes B, B causes A, or some unknown factor effects both A and B.
To calculate the correlation coefficient, we need to first compute the mean and deviation scores for both the dexterity and anxiety variables.
Let's start by calculating the mean for each variable. Add up all the dexterity scores (1 + 1 + 2 + 4) and divide by the number of participants (4). The mean dexterity score is 2. Similarly, add up all the anxiety scores (10 + 8 + 4 -2) and divide by 4 to get a mean anxiety score of 5.
Next, we need to calculate the deviation scores for each participant. To find the deviation score for dexterity, subtract the mean dexterity score (2) from each participant's dexterity score. For participant 1, the deviation score would be 1 - 2 = -1. Repeat this process for all participants. Similarly, find the deviation scores for anxiety by subtracting the mean anxiety score (5) from each participant's anxiety score.
Now we have the mean and deviation scores for both variables. Multiply each participant's deviation score for dexterity by their deviation score for anxiety. For participant 1, -1 × 5 = -5. Repeat this process for all participants.
Add up all the products you obtained from the previous step (-5 + -3 + -2 + 8) to get the sum of the products (SP). SP = -2.
To calculate the correlation coefficient, we need to use the following formula:
r = SP / (√sum of squared deviations for dexterity × √sum of squared deviations for anxiety)
To find the sum of squared deviations for dexterity, square each deviation score for dexterity and add them up. For participant 1, (-1)^2 = 1. Repeat this process for all participants. Then find the sum of these squared deviations.
Similarly, square each deviation score for anxiety and add them up to find the sum of squared deviations for anxiety.
Let's say the sum of squared deviations for dexterity is 10 and the sum of squared deviations for anxiety is 30.
Now put the values in the formula:
r = -2 / (√10 × √30)
After solving this equation, we find that the correlation coefficient (r) is approximately -0.377.
To determine whether the correlation is statistically significant, we compare the calculated correlation coefficient to the critical value from a statistical table. In this case, we need to use a significance level of 0.05 (two-tailed) since we want to check for significance in both positive and negative directions.
Since we have only four participants, it is not appropriate to use conventional statistical tests like the t-test. Instead, we can use a table specifically designed for small sample sizes. Consulting this table, we find that a correlation coefficient of -0.377 is not statistically significant at the 0.05 level.
Now, let's explain the logic behind the steps we took. Correlation measures the degree of association between two variables, in this case, dexterity and anxiety. By calculating the correlation coefficient, we can determine the strength and direction of the relationship between the two variables. The correlation coefficient ranges from -1 to +1, where -1 indicates a perfect negative correlation, +1 indicates a perfect positive correlation, and 0 indicates no correlation.
To compute the correlation coefficient, we first standardized the data by calculating deviation scores. Deviation scores represent the distance of each data point from the mean. By standardizing the data, we eliminate the influence of the scale and bring the variables to a common metric.
The correlation coefficient formula calculates the relationship between the deviations of the two variables. The numerator, the sum of the products, determines the direction of the correlation (whether it is positive or negative), and the denominator represents their dispersion or variation.
To determine if the correlation is statistically significant, we compare the calculated correlation coefficient to the critical value. This critical value corresponds to the level of significance, which is typically set at 0.05. If the calculated correlation coefficient falls within the critical region, we can conclude that the correlation is statistically significant.
Lastly, let's consider three possible directions of causality for the correlation in light of the variables involved:
1. Dexterity causing anxiety: It is reasonable to assume that individuals with better manual dexterity might experience less anxiety when performing tasks that require manual skills. This is because increased proficiency and confidence in manual tasks can reduce anxiety.
2. Anxiety causing dexterity: It is also possible that individuals with higher anxiety levels might exhibit poorer manual dexterity due to the physical and cognitive effects of anxiety. Increased stress and tension could negatively impact fine motor skills, leading to lower dexterity scores.
3. Shared underlying factors: Another plausible explanation is that both dexterity and anxiety might be influenced by common factors such as genetics, personality traits, or neurological factors. These shared factors could contribute to the observed correlation, without one variable directly causing the other.
Keep in mind that while the correlation coefficient provides information about the strength and direction of the relationship between variables, it does not indicate causation. Correlation simply highlights the association between variables, and additional research is necessary to establish causal relationships.