Question # 1 A well-known multinational FMCG produces more than fifty products and sells them in the market. Out of fifty products one is mineral water for drinking purpose and supplies different size pack bottles in the market. Product manager observed that, during hot days the sale of water is more as compared to the normal days. He started the collection of data of weekly sale of water and the average weekly temperature. Product manager was interested in developing a relationship between sales and temperature; therefore, he discussed and shared the following data with you. sales 75 77 69 63 79 80 77 70 73 87 72 73 80 80 69 84 77 72 74 81 temperature 32 36 31 31 34 29 32 35 28 31 30 31 29 35 36 28 32 33 32 33 Sale is in thousand liters and temperature is in degree Celsius. Data is Normally distributed You are required to determine the following information for some concrete decision: Determine the statistical relationship between temperature and sale of water and interpret. How much change in sales if the temperature increases by two degrees? If for instance, the temperature becomes zero then how much sales will be? How much percent variation in sales due to temperature and due to other factors? Write the equation for the prediction of sales.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter4: Equations Of Linear Functions

Section4.5: Correlation And Causation

Problem 15PPS

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Question # 1

A well-known multinational FMCG produces more than fifty products and sells them in the market. Out of fifty products one is mineral water for drinking purpose and supplies different size pack bottles in the market. Product manager observed that, during hot days the sale of water is more as compared to the normal days. He started the collection of data of weekly sale of water and the average weekly temperature.

Product manager was interested in developing a relationship between sales and temperature; therefore, he discussed and shared the following data with you.

sales	75	77	69	63	79	80	77	70	73	87	72	73	80	80	69	84	77	72	74	81
temperature	32	36	31	31	34	29	32	35	28	31	30	31	29	35	36	28	32	33	32	33

Sale is in thousand liters and temperature is in degree Celsius. Data is Normally distributed

You are required to determine the following information for some concrete decision:

Determine the statistical relationship between temperature and sale of water and interpret.
How much change in sales if the temperature increases by two degrees?
If for instance, the temperature becomes zero then how much sales will be?
How much percent variation in sales due to temperature and due to other factors?
Write the equation for the prediction of sales.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Expert Solution

Step 1

“Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for you. To get remaining sub-part solved please repost the complete question and mention the sub-parts to be solved.”

Let (X,Y) be the bivariate distribution where Y is the dependent and X is the independent variable. The line of regression is the line which gives the best estimate to the values of one variable for the specific value of the other variable. The line of regression is obtained by the principle of least square. The equation of the line of regression of Y on X is given as:

$Y = a + b X$

Here, a and b are constants obtained as:

$\begin{array}{rcl} a & = & \frac{\sum_{i = 1}^{n} y_{i}}{n} \\ b & = & \frac{{\sum x_{i}}_{i = 1}^{n} y_{i}}{{\sum {x_{i}}^{2}}_{i = 1}^{n}} \end{array}$

Step 2

(1.) Let X denotes temperature and Y denotes sales. Here, $n = 20$ .

Calculation Table:

x_i	y_i	x_i²	x_iy_i
32	75	1024	2400
36	77	1296	2772
31	69	961	2139
31	63	961	1953
34	79	1156	2686
29	80	841	2320
32	77	1024	2464
35	70	1225	2450
28	73	784	2044
31	87	961	2697
30	72	900	2160
31	73	961	2263
29	80	841	2320
35	80	1225	2800
36	69	1296	2484
28	84	784	2352
32	77	1024	2464
33	72	1089	2376
32	74	1024	2368
33	81	1089	2673
Total	1512	20466	48185

Step 3

Using above calculation table, we get

$\begin{array}{rcl} a & = & \frac{\sum_{i = 1}^{n} y_{i}}{n} \\ = & \frac{1512}{20} \\ = & 75.6 \\ b & = & \frac{{\sum x_{i}}_{i = 1}^{n} y_{i}}{{\sum {x_{i}}^{2}}_{i = 1}^{n}} \\ = & \frac{48185}{20466} \\ = & 2.3544 \end{array}$

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