Question

a) It is suggested that the shell thickness of hens' eggs increases with the amount of grit that is added to their food. Eight hens were given varying amounts of grit (x [in grams]) in their food and the shell thickness (y [in tenths of a millimetre]) of an egg laid by each hen a month later was measured. The results can be summarised as follows: Ex = 216; Ey=48; Σ.x2 = 6672; E xy = 1438. i. Find sand Sxy. ii. Find the equation of the regression line of y on x. iii. Use your equation found in part ii to estimate the shell thickness of an egg laid by a hen which has 15 grams of grit added to the food. The masses of the eggs laid by the hens can be assumed to follow a Normal distribution with mean 54 grams and standard deviation 5 grams. An egg is classified as 'medium' if its mass lies between 48 grams and 60 grams. iv. Find the percentage of eggs which are 'medium'. The eggs are packed in trays of 30. V. Find the probability that a tray selected at random has exactly 25 or exactly 26 'medium' eggs. [2] [2] [2] [5] [3]

Answer 1

a) The value of Sxy is -1154. b) The equation of the regression line of y on x is y = 6.5875 - 0.1732x. c) The estimated shell thickness of an egg with 15 grams of grit is 3.8805 tenths of a millimetre. d) The percentage of eggs which are 'medium' can be calculated using the normal distribution and z-scores. e) The probability that a tray selected at random has exactly 25 or exactly 26 'medium' eggs can be calculated using the binomial distribution.

Step-by-step explanation:

We are given the following information:
Ex = 216
Ey = 48
Σx^2 = 6672
E_xy = 1438

i. To find the value of Sxy, we can use the formula Sxy = E_xy - Ex * Ey / n, where n is the number of observations. Plugging in the given values, we get Sxy = 1438 - 216 * 48 / 8 = 1438 - 2592 = -1154.

ii. The equation of the regression line of y on x can be found using the formula y = a + bx, where b = Sxy / Sx^2 and a = Ey / n - b * Ex / n. Plugging in the values, we get b = -1154 / 6672 and a = 48 / 8 - (-1154 / 6672) * 216 / 8. Simplifying, we find b = -0.1732 and a = 6.5875. Therefore, the equation is y = 6.5875 - 0.1732x.

iii. To estimate the shell thickness of an egg with 15 grams of grit, we can substitute x = 15 into the equation found in part ii. Using the equation y = 6.5875 - 0.1732x, we find y = 6.5875 - 0.1732 * 15 = 3.8805. Therefore, the estimated shell thickness is 3.8805 tenths of a millimetre.

iv. To find the percentage of 'medium' eggs, we need to determine the number of eggs with a mass between 48 grams and 60 grams. We can use the normal distribution and z-scores to find this probability. First, we calculate the z-scores for both values: z1 = (48 - 54) / 5 and z2 = (60 - 54) / 5. Using z-tables or a calculator, we find the corresponding probabilities of the z-scores. Subtracting these probabilities, we get the percentage of 'medium' eggs.

v. To find the probability that a tray selected at random has exactly 25 or exactly 26 'medium' eggs, we can use the binomial distribution. The probability of getting exactly k 'medium' eggs out of 30 trays can be calculated using the formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of 'medium' eggs, and p is the probability of success for one trial. Plugging in the values, we can calculate P(X = 25) + P(X = 26) for k = 25 and k = 26, respectively.

Answer 2

i.
`S_(xy)=142`for the given data.

ii. Calculate r, use it in the regression line equation; then estimate y when x=15.

iii. Utilize Normal distribution parameters to find the percentage of 'medium' eggs and the probability of 25 or 26 'medium' eggs in a tray of 30.

i. Find and Calculate Sxy:

Using the given information, we have:

`E_x = 216, E_y = 48, \Sigma x^2 = 6672, E _(xy )= 1438`

The formula for Sxy is:

`S_(xy) = E_((xy)) - (E_x)(E_y)/n`

where n is the number of data points.

Plugging in the values:

`S_(xy) = 1438 - (216 * 48)/8`

`S_(xy) = 1438 - 10368/8`

`S_(xy) = 1438 - 1296`

`S_(xy) = 142`

Therefore,
`S_(xy) is 142`.

ii. Find the equation of the regression line:

The equation of the regression line is given by:

y = a + bx

where a is the y-intercept and b is the slope of the line.

The formulas for a and b are:

`b = S_(xy) / S_(xx)`

`a = (E_y / n) - (b * (E_x /n))`

where Sxx is the sum of squares of x and n is the number of data points.

Plugging in the values:

Sₓₓ = Σx² - (Eₓ² / n)

Sₓₓ = 6672 - (216² / 8)

Sₓₓ = 6672 - 5832

Sₓₓ = 840

b = 142/840

b = 0.169

a = (48/8) - (0.169 × (216/8))

a = 6 - (0.169 × 27)

a = 6 - 4.563

a = 1.437

Therefore, the equation of the regression line is y = 1.437 + 0.169x.

iii. Use the equation to estimate the shell thickness of an egg with 15 grams of grit:

Using the equation y = 1.437 + 0.169x, we can substitute x with 15:

y = 1.437 + (0.169 × 15)

y = 1.437 + 2.535

y = 3.972

Therefore, the estimated shell thickness of an egg with 15 grams of grit is 3.972 tenths of a millimeter.