Question

3. Dr. Ayodele assessed the correlation between exam grade and time allowed in a biostatistics test. The test result showed a Pearson correlation value of 1.09. Interpret this r-value in terms of strength and direction.

4. Using a set of data collected over a period of one year, a researcher assessed the linear relationship between urban growth and air quality violations in a community. The assessment showed a Pearson correlation coefficient(r) of 0.88 between urban growth and air quality violation. The significance was tested at 0.01 and showed a p value of 0.002. Explain this relationship in terms of strength, direction, and significance.

5. In a study conducted by Dr. Ayodele, to examine the association between cigarette smoking and low birth weight, the findings showed a very strong correlation between cigarette smoking and low birth weight. Dr. Ayodele concluded that the low birth weight observed in the study was caused by cigarette smoking. Critically review the conclusion of this study and comment.

6. The test results of a correlation analysis conducted to assess the relationship between body mass index and body fat percentage indicated a Pearson correlation coefficient (r) value of 0.71. Interpret the r value in terms of strength and direction. The significance (p= 0.14) was tested at α = 0.01. Was there a statistically significant correlation between body mass index and body fat percentage? How did you arrive at your conclusion?

7. Assume that you have paired values consisting of heights(in inches) and weights(in lb.) from 40 randomly selected men. The linear correlation coefficient r is 0.4500. Find the value of the coefficient of determination.

8. The method of one-way analysis of variance is used for tests of hypotheses that three or more population means are equal. Yes or No?

9. Twenty different statistics students are randomly selected. For each of them, their body temperature (degrees°C) is measured and their head circumference(cm) is measured. If it is found that r equals=0, does that indicate that there is no association between these two variables?

10. Linear correlation coefficient r measures the strength of the linear correlation between the paired quantitative x- and y-values in a sample. Yes or No?

11. Assume that you have paired values consisting of heights(in inches) and weights(in lb) from 40 randomly selected men. The linear correlation coefficient r is 0.4390. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

12. Use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between bear neck size and weight (x equals=neck size, y equals=weight). r equals=0.2040.What is the value of the coefficient of determination? What is the percentage of the total variation that can be explained by the linear relationship between bear neck size and weight?

Answer 1

Before Answering the question i would like to mention a few things related to these questions which will be used for better understanding of the answers.

1)Null hypothesis H0 is that population correlation coefficient =0 and alternative hypothesis H1  is that population correlation coefficient is significantly different from 0.

2) on the basis of p value we may reject the null hypothesis at alpha level of significance if p values is less than alpha otherwise we fail to reject null hypothesis due to lack of evidence.(Note that we never accept the null hypothesis we either reject it or fail to reject)

3) pearson correlation cofficient measures the strength and direction(nature) of linear relationship between 2 variables, whereas coefficient of determination measures measures the pearson correlation between a dependent variable and the linear function of remaining independent or explanatory variables.

now lets start answering the questions.

#3) range of the pearson correlation coefficient lies between -1 to 1, hence it cant take the value 1.09 which means there must be some calculation error while computing it.

#4) correlation coefficient of .88 implies there is strong positive correlation between urban growth and air quality violation, i.e for 100 units increase (or decrease) in 1 variable the other will increase (or decrease) by 88 units. when both these variables will be plotted there will be an almost straight line. moreover from the concept of p value discussed above we may reject the null hypothesis at 1% level of significance as p values is less than 0.01. which means we 99% times our null hyposthesis will be rejected.

from the point 1 mentioned above and rejecting the null hypothesis at 1% level of significance we may conclude that population correlation is significantly different from 0 and this correlation value=.88 is significant.

#5) first of all Dr. Ayodele's conclusion is based on a sample only and it is not appropriate to generalise it for the whole populaion without checking the significance of correlation value obtained from sample. moreover birth weight is determined by many other factors like nutrition of mother,environment,etc so concluding the result just on the basis of one factor is not right.

#6) 0.71 correlation coefficient between body mass index and body fat percentage means with increase in BMI there is also increase in body fat percentage.if we try to establish a linear relationship between these 2 variables only 29% values will deviate from fitted line. since p value=.14 we fail to reject null hypothesis at 1%,5% and even 10% level of significance. so we may conclude that the correlation coefficient=.71 is not significant and is due to the chosen sample only and not true for whole population.

#7)coefficient of determination=R2=r2 =0.4500*0.4500=0.25.

#8) Yes Analysis of Variance or ANOVA technique is used to check equalitu of means when we have more than 2 samples.

#9) correlation=0 means that there is no relationship between 2 variables, it should not be confused with association there can be many other non linear relationships between variables which are not measured by correlation coefficient.

#10) YES

#11) R2=r2 = .4390*.4390=.19. coefficient of determination is measures how well the linear function of independent variables explains the dependent variable.R2=.19 is considered very low.

#12)R2=r2 =.2040*.2040=.0416. this means about 4% of variation in weight is explained by size of neck