6.03 Correlation coefficient
Describe the type of linear relationship between the variables in the following scatter plots:
Estimate the value of the correlation coefficient for the data in the following scatter plots:
Describe the linear relationship between the variables with the following correlation coefficients:
0.96
0.66
0.36
- 0.06
- 0.34
- 0.66
If the explanatory variable increases, describe the effect on the response variable for the following studies:
A study found that the correlation coefficient between heights of women and probability of being turned down for a promotion was found to be - 0.90.
A study found that the correlation coefficient between population of a city and number of speeding fines recorded was found to be 0.83.
A study found that the correlation coefficient between length of hair and length of fingernails was found to be 0.07.
A study found that the correlation coefficient between number of bylaws a council has about dog breeding and number of dogs available for adoption at the local shelter was found to be 0.55.
A pair of data sets have a correlation coefficient of \dfrac{1}{10} while a second pair of data sets has a correlation coefficient of \dfrac{3}{5}. Which pair of data sets have the stronger correlation?
Describe the type of correlation the following correlation coefficients indicate:
r = 1
r = 0
r = -1
For each of the following graphs, write down an estimate of the correlation coefficient:
For each of the following pairs of relationships:
Determine whether the two linear relationships have the same direction.
State the relationship that has the stronger correlation.
The linear relationship between a set of data for variables x and y has a correlation coefficient of 0.3. The linear relationship between a set of data for variables x and z has a correlation coefficient of 0.9.
The linear relationship between a set of data for variables x and y has a correlation coefficient of - 0.9. The linear relationship between a set of data for variables y and z has a correlation coefficient of - 0.5.
The linear relationship between a set of data for variables x and y has a correlation coefficient of 0.8. The linear relationship between a set of data for variables s and t has a correlation coefficient of - 0.5.
The linear relationship between a set of data for variables x and y has a correlation coefficient of 0.3. The linear relationship between a set of data for variables x and t has a correlation coefficient of - 0.9.
A researcher plotted the life expectancy of a group of men against the number of cigarettes they smoke a day. The results were recorded and the correlation coefficient r was found to be - 0.88.
Describe the correlation between the life expectancy of a man and the number of cigarettes smoked per day.
A researcher was evaluating the relationship between the number of years in education a person completes and the number of pets they own. The results were recorded and correlation coefficient r was found to be - 0.3.
Describe the correlation between a person's years of education and the number of pets they own.
The scatter diagram shows data of the height of a ball kicked into the air as a function of time:
Which type of model is appropriate for the data, linear or non-linear?
Write down a possible value of Pearson’s correlation coefficient, r, for this set of data.
The scatter diagram shows data of a person's level of happiness as a function of their age:
Which type of model is appropriate for the data, linear or non-linear?
Write down a possible value of Pearson’s correlation coefficient, r, for this set of data.
The scatter diagram shows data of the height of an object after it is pushed off a rooftop as a function of time:
Which type of model is appropriate for the data, linear or quadratic?
Write down a possible value of Pearson’s correlation coefficient, r, for this set of data.
Determine whether the following describe a relationship that is correlated but not causal:
The sales of ice cream and increase in temperature.
The number of hours worked and how much money is made for a given person.
The amount of showers had in a day and the amount of the water bill.
The amount of rainfall received, and level of water in a lake.
The larger the dimensions of a rectangular verandah, the more area.
The season of the year and the number of water related injuries.
Increase in temperature, and the level of mercury in a thermometer.
The number of students shouting in class and the number of detentions received.
For each of the following data examples, determine if there is a causal relationship between the variables:
The number of times a coin lands on heads and the likelihood that it lands on heads on the next flip.
The amount of weight training a person does and their strength.
Determine whether the following describe a causal relationship and not just a correlation:
An individual's decision to work in construction and his diagnosis of skin cancer.
The number of minutes spent exercising and the amount of calories burned.
A decrease in temperature and the increase in attendance at an ice skating rink.
As a child's weight increases, so does her vocabulary.
Determine whether the following are examples of variables with no correlation:
The age of a child and their shoe size.
The age of a child and their height.
The age of a child and the number of pets owned.
The age of a child and the amount of adjectives learned.
The table shows the number of fans sold at a store during days of various temperatures:
| \text{Temperature (\degree C)} | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
|---|---|---|---|---|---|---|---|---|
| \text{Number of fans sold} | 12 | 13 | 14 | 17 | 18 | 19 | 21 | 23 |
For this data, will r be greater than zero, less than zero or equal to zero?
Is there a causal relationship between the variables? Explain your answer.
A study found a strong positive association between the temperature and the number of beach drownings.
Does this mean that the temperature causes people to drown? Explain your answer.
Is the strong correlation found a coincidence? Explain your answer.
Many trees lose their leaves in winter. Does this mean that cold temperatures cause the leaves to fall?
A study found a strong correlation between the approximate number of pirates out at sea and the average world temperature.
Does this mean that the number of pirates out at sea has an impact on world temperature?
Is the strong correlation found a coincidence? Explain your answer.
If there is correlation between two variables, is there causation?
Scatter plots for two sets of data are shown below:
Set A
Set B
Which data set has the strongest linear correlation between the variables?
Which data set appears to have a non-linear relationship? Explain your answer.
Why does data set B have the weakest correlation between its variables?
State whether the following pairs of variables could be represented by data set B:
Marks in an English examination and distance travelled from home to school.
Cost of cars and cost of petrol.
Distance travelled in a car and the cost of a driver’s license.
Height and weight of students at school.