Answer :
We are given the following data over six years:
[tex]$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Price in Rs. per kg} & 100 & 110 & 112 & 115 & 120 & 140 \\
\hline
\text{Supply in kg} & 30 & 40 & 45 & 20 & 55 & 55 \\
\hline
\end{array}
$$[/tex]
We wish to:
(a) Find the coefficient of correlation between price and supply.
(b) Estimate the supply (in kg) when the price is Rs. 150 using linear regression.
Below is a detailed solution.
────────────────────────────
Step 1. Calculate the Coefficient of Correlation
The coefficient of correlation (usually denoted by [tex]$r$[/tex]) provides a measure of the strength and direction of the linear relationship between two variables. After computing using the given data, we obtain:
[tex]$$
r \approx 0.6193.
$$[/tex]
This indicates a moderately positive linear relationship between price and supply.
────────────────────────────
Step 2. Form the Linear Regression Model
To estimate the supply when the price is Rs. 150, we use the linear regression model of the form:
[tex]$$
\text{Supply} = m \times (\text{Price}) + c,
$$[/tex]
where [tex]$m$[/tex] is the slope and [tex]$c$[/tex] is the intercept.
From the analysis of the data, the slope and intercept are found to be approximately:
[tex]$$
m \approx 0.6429 \quad \text{and} \quad c \approx -33.8529.
$$[/tex]
Thus, the regression equation becomes:
[tex]$$
\text{Supply} \approx 0.6429 \times (\text{Price}) - 33.8529.
$$[/tex]
────────────────────────────
Step 3. Estimate the Supply at Rs. 150
Substitute the value [tex]$\text{Price} = 150$[/tex] into the regression equation:
[tex]$$
\text{Supply} \approx 0.6429 \times 150 - 33.8529.
$$[/tex]
Calculate the above expression step-by-step:
1. Multiply the slope by 150:
[tex]$$
0.6429 \times 150 \approx 96.435.
$$[/tex]
2. Subtract the intercept:
[tex]$$
96.435 - 33.8529 \approx 62.5851.
$$[/tex]
Thus, when the price is Rs. 150, the estimated supply is approximately
[tex]$$
62.59 \text{ kg} \quad (\text{rounded to two decimal places}).
$$[/tex]
────────────────────────────
Final Answers:
(a) The coefficient of correlation between price and supply is approximately [tex]$0.6193$[/tex].
(b) The estimated supply when the price is Rs. 150 is approximately [tex]$62.59$[/tex] kg.
[tex]$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Price in Rs. per kg} & 100 & 110 & 112 & 115 & 120 & 140 \\
\hline
\text{Supply in kg} & 30 & 40 & 45 & 20 & 55 & 55 \\
\hline
\end{array}
$$[/tex]
We wish to:
(a) Find the coefficient of correlation between price and supply.
(b) Estimate the supply (in kg) when the price is Rs. 150 using linear regression.
Below is a detailed solution.
────────────────────────────
Step 1. Calculate the Coefficient of Correlation
The coefficient of correlation (usually denoted by [tex]$r$[/tex]) provides a measure of the strength and direction of the linear relationship between two variables. After computing using the given data, we obtain:
[tex]$$
r \approx 0.6193.
$$[/tex]
This indicates a moderately positive linear relationship between price and supply.
────────────────────────────
Step 2. Form the Linear Regression Model
To estimate the supply when the price is Rs. 150, we use the linear regression model of the form:
[tex]$$
\text{Supply} = m \times (\text{Price}) + c,
$$[/tex]
where [tex]$m$[/tex] is the slope and [tex]$c$[/tex] is the intercept.
From the analysis of the data, the slope and intercept are found to be approximately:
[tex]$$
m \approx 0.6429 \quad \text{and} \quad c \approx -33.8529.
$$[/tex]
Thus, the regression equation becomes:
[tex]$$
\text{Supply} \approx 0.6429 \times (\text{Price}) - 33.8529.
$$[/tex]
────────────────────────────
Step 3. Estimate the Supply at Rs. 150
Substitute the value [tex]$\text{Price} = 150$[/tex] into the regression equation:
[tex]$$
\text{Supply} \approx 0.6429 \times 150 - 33.8529.
$$[/tex]
Calculate the above expression step-by-step:
1. Multiply the slope by 150:
[tex]$$
0.6429 \times 150 \approx 96.435.
$$[/tex]
2. Subtract the intercept:
[tex]$$
96.435 - 33.8529 \approx 62.5851.
$$[/tex]
Thus, when the price is Rs. 150, the estimated supply is approximately
[tex]$$
62.59 \text{ kg} \quad (\text{rounded to two decimal places}).
$$[/tex]
────────────────────────────
Final Answers:
(a) The coefficient of correlation between price and supply is approximately [tex]$0.6193$[/tex].
(b) The estimated supply when the price is Rs. 150 is approximately [tex]$62.59$[/tex] kg.
