I see that you’ve uploaded an image containing two questions related to correlation and regression. Let me analyze and provide a detailed response:

1. Define correlation and explain its significance. Discuss the different types of correlation with the help of scatter plots and numerical examples (a)?

Part (a):

Given the data:

• $y$: Number of grains per panicle
  $112, 131, 147, 90, 110, 106, 127, 145, 85, 94, 142, 111$
• $x$: Length of panicles in cm
  $22.9, 23.9, 24.8, 21.2, 22.2, 22.7, 23.0, 24.0, 20.6, 21.0, 24.0, 23.1$

You need to calculate the correlation coefficient ($r$) and perform a significance test.

The correlation coefficient formula is: $r = \frac{n \sum xy - \sum x \sum y}{\sqrt{\left(n \sum x^2 - (\sum x)^2\right)\left(n \sum y^2 - (\sum y)^2\right)}}$

Where:

• $n$: Number of observations
• $x, y$: Individual values of the variables

The steps to solve:

1. Compute $\sum x, \sum y, \sum xy, \sum x^2, \sum y^2$.
2. Substitute the values into the correlation formula.
3. Compare $r$ with the critical $t$-value ($t_{0.05, 10} = 2.23$) to determine statistical significance.

2. Define regression and explain its properties. Differentiate between correlation and regression and numerical examples (b)?

Part (b):

Given data:

• $x$: Days to flowering
  $91, 76, 98, 136, 128, 119, 142, 135, 147, 145, 122$
• $y$: Yield per plant (gm)
  $24.72, 20.25, 38.56, 74.72, 72.75, 78.45, 69.8, 80.4, 160.2, 165.75, 77.56$

You need to:

1. Fit regression equations:
   • $Y$ on $X$: $Y = a + bX$
   • $X$ on $Y$: $X = a' + b'Y$
2. Estimate $Y$ when $X = 149$.

The regression equation $Y = a + bX$ requires: $b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}, \quad a = \frac{\sum y}{n} - b \frac{\sum x}{n}$

For $X = a' + b'Y$, a similar calculation applies.

Finally, substitute $X = 149$ into the regression equation of $Y$ on $X$ to estimate $Y$.

Next Steps:

Let me know if you’d like me to perform the detailed calculations for either question.

Related Questions:

1. What are the assumptions of correlation and regression analysis?
2. How do scatter plots help visualize the relationship between two variables?
3. What is the difference between a positive and a negative correlation?
4. How can you interpret the slope in a regression equation?
5. What are the real-world applications of correlation and regression?

Tip:

Always check the range of data for outliers before computing correlation or regression, as outliers can distort the results significantly.