The matric class of a certain high school had to vote for the chairperson of the RCL (representative council of learners) - NSC Mathematics - Question 1 - 2022 - Paper 2

To calculate the standard deviation (SD), we first find the differences from the mean, square them, sum these squares, and then divide by the number of learners:

1. First, compute each difference from the mean:
   • (9 - 15.5)² = 42.25
   • (22 - 15.5)² = 42.25
   • (10 - 15.5)² = 30.25
   • (21 - 15.5)² = 30.25
   • (11 - 15.5)² = 20.25
   • (15 - 15.5)² = 0.25
   • (20 - 15.5)² = 20.25
   • (12 - 15.5)² = 12.25
   • (19 - 15.5)² = 12.25
   • (16 - 15.5)² = 0.25
2. The sum of squared differences = 42.25 + 42.25 + 30.25 + 30.25 + 20.25 + 0.25 + 20.25 + 12.25 + 12.25 + 0.25 = 206.5
3. Now, calculate the variance and then take the square root for standard deviation:
   SD = rac{206.5}{10} = 20.65 \ SD = ext{sqrt}(20.65) ext{ which approximately equals } 4.59
   Thus, the standard deviation is approximately 4.59.

Those who received fewer votes than one standard deviation below the mean are considered not invited. Therefore, we find:
$ext{Mean} - SD = 15.5 - 4.59 = 10.91$
Since 10.91 is the threshold, we check how many learners received fewer votes:

• Votes received: 9, 10, 11
  The learners with votes of 9, 10, and 11 were not invited:
  Thus, 7 learners were invited.

The least squares regression line can be determined using the formula:
$y = a + bx$

1. Calculate the coefficients:
   • $a = 1.7709$
   • $b = 0.2243$
2. Therefore, the equation of the regression line is:
   $y = 1.77 + 0.22x$
   This can be approximated as y = 1.77 + 0.22x.

Using the equation from previous calculations:
Substituting x = 72 into the regression equation:
$y = 1.77 + 0.22(72)$
Calculating:
$y = 1.77 + 15.84 = 17.61$
Thus, the predicted number of votes is approximately 18 votes.

The scatter plot indicates that there is a low correlation between IQ and the number of votes received. Most points are scattered without a discernible trend, suggesting that IQ does not strongly influence voting outcomes.

The prediction in QUESTION 1.4 is reliable because the correlation coefficient is relatively strong, with a calculated value of approximately 0.98. This indicates a strong relationship between the independent variable (popularity score) and the dependent variable (number of votes). Predictions fall within a reasonable range based on the data set.