Question

Measures of correlation are used for identifying relationships between different types of variables. The most appropriate measure of correlation to use for any given data set, depends on the type of variables and levels of measurement.

Which is best?

For this discussion, begin by creating an example of a correlational study. Consider what variables it would analyze and what type of data would be collected. Then, develop a main response in which you address the following:

• Briefly identify the topic and purpose of your example.
• Describe the variables and how they would be measured.
• Explain which measure of correlation would be most appropriate.
• Discuss any limitations of the correlational approach that are relevant.

Post your main response in accordance with the time frame outlined below, and then reply to the main responses of at least two other students.

Answer 1

Solution:

1) Ice Cream Sales Problem:

We would like to estimate the relationship between the sales of ice-creams with rising temperature. for example, the local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for the last 12 days which he has recorded. The data set for the problem is given below

Ice Cream Sales vs Temperature
Temperature °C	Ice Cream Sales
14.2°	$215
16.4°	$325
11.9°	$185
15.2°	$332
18.5°	$406
22.1°	$522
19.4°	$412
25.1°	$614
23.4°	$544
18.1°	$421
22.6°	$445
17.2°	$408

2)

The two measurable variable that will be included in the analysis would be a) Temperature °C and b) Ice Cream Sales.

3)

Pearson correlation of coefficient would be better to estimate degree of relationship between the two variables. which is estimated via the following formula given below.

$r = \frac{n\sum x_{i}y_{i}-\sum x_{i} \sum y_{i}}{\sqrt{n\sum x_{i}^{2}-(\sum x_{i})^{2}} \sqrt{n\sum y_{i}^{2}-(\sum y_{i})^{2}}}$ where X and Y are two variables

Scatter plot for the two variables is given below:

Scatterplot of Ice Cream Sales (y) vs Temperature °C (x) 600 500 Ice Cream Sales (y) 400 300 200 10 12 14 22 24 26 16 18 20 Temperature °C (x)

The calculation for the same is given below in the table

Temperature °C (x)	Ice Cream Sales (y)	XY	X2	Y2
14.2	215	3053	201.64	46225
16.4	325	5330	268.96	105625
11.9	185	2201.5	141.61	34225
15.2	332	5046.4	231.04	110224
18.5	406	7511	342.25	164836
22.1	522	11536.2	488.41	272484
19.4	412	7992.8	376.36	169744
25.1	614	15411.4	630.01	376996
23.4	544	12729.6	547.56	295936
18.1	421	7620.1	327.61	177241
22.6	445	10057	510.76	198025
17.2	408	7017.6	295.84	166464
$\sum x_{i} =$ 224.1	$\sum y_{i} =$ 4829	$\sum x_{i} y_{i} =$ 95506.6	$\sum x_{i}^{2} =$ 4362.05	$\sum y_{i}^{2} =$  2118025

  $r = \frac{n\sum x_{i}y_{i}-\sum x_{i} \sum y_{i}}{\sqrt{n\sum x_{i}^{2}-(\sum x_{i})^{2}} \sqrt{n\sum y_{i}^{2}-(\sum y_{i})^{2}}}$

$r = 0.9575$

4)

The pearson correlation coefficient is valid only if the relationship is linear between the variables. There is the linear relationship between Temperature and Ice cream sales as given in above scatterplot.

Please give upvote.

Thank you.