(i) Let the cost of a sweater be \(x\).

The total cost of the items is given by:

\(4 \times 5.5 + 3x + 90 = 8 \times 29\)

\(22 + 3x + 90 = 232\)

\(112 + 3x = 232\)

\(3x = 120\)

\(x = 40\)

Thus, the cost of a sweater is $40.

(ii) Since the standard deviation is $0, all items cost the same. Therefore, the cost of each item is $26. Diksha buys 4 shirts, so the total cost for the shirts is:

\(4 \times 26 = 104\)

Thus, Diksha spends $104 on shirts.

To estimate the mean, we first find the midpoints of each class interval:

• 1 – 25: midpoint = 13
• 26 – 35: midpoint = 30.5
• 36 – 45: midpoint = 40.5
• 46 – 55: midpoint = 50.5
• 56 – 90: midpoint = 73

Next, we calculate the mean using the formula:

\(\bar{x} = \frac{\sum f x}{\sum f} = \frac{4 \times 13 + 24 \times 30.5 + 38 \times 40.5 + 34 \times 50.5 + 20 \times 73}{120} = \frac{5500}{120} = 45.8\)

\(To find the variance, we use:\)

\(s^2 = \frac{\sum f x^2}{\sum f} - \bar{x}^2\)

\(\sum f x^2 = 4 \times 13^2 + 24 \times 30.5^2 + 38 \times 40.5^2 + 34 \times 50.5^2 + 20 \times 73^2 = 278620\)

\(s^2 = \frac{278620}{120} - 45.8^2 = 2321.8333 - 2097.64 = 224.1933\)

The standard deviation is:

\(s = \sqrt{224.1933} = 14.9\)

(i) To find the mean weight of the 27 birds, use the formula for the combined mean:

\(\text{new mean} = \frac{9 \times 7.1 + 18 \times 5.2}{27} = 5.83 \text{ kg}\)

(ii) First, find \(\Sigma x_t^2\) and \(\Sigma x_g^2\) using the variance formula:

For turkeys: \(1.45^2 = \frac{\Sigma x_t^2}{9} \Rightarrow \Sigma x_t^2 = 472.6125\)

For geese: \(0.96^2 = \frac{\Sigma x_g^2}{18} - 5.2^2 \Rightarrow \Sigma x_g^2 = 503.3088\)

Now, find the combined standard deviation:

\(\text{New } s^2 = \frac{472.6125 + 503.3088}{27} - 5.83^2 = 2.117\)

\(\text{New } s = \sqrt{2.117} = 1.46 \text{ kg}\)

To find the mean, sum all the data points and divide by the number of data points:

\(\text{Mean} = \frac{5 + (-2) + 12 + 7 + (-3) + 2 + (-6) + 4 + 0 + 8}{10} = \frac{27}{10} = 2.7\)

\(To find the variance, use the formula:\)

\(\text{Variance} = \frac{(5^2 + (-2)^2 + 12^2 + 7^2 + (-3)^2 + 2^2 + (-6)^2 + 4^2 + 0^2 + 8^2)}{10} - (\text{Mean})^2\)

\(= \frac{35.1}{10} - 2.7^2 = 3.51 - 7.29 = 27.8\)

(i) To find the mean height of the remaining group of 27 people, we use the formula for the mean:

\(\text{New mean} = \frac{172.6 \times 28 - 161.8}{27} = 173 \text{ cm}\)

(ii) First, find \(\Sigma x^2\) for the original group of 28 people using the formula:

\(\Sigma x^2 = (4.58^2 + 172.6^2) \times 28 = 834728.6\)

Then, for the remaining group of 27 people:

\(\text{Remaining } \Sigma x^2 = 834728.6 - 161.8^2 = 808549.36\)

Finally, calculate the standard deviation for the remaining group:

\(\text{sd of remaining} = \sqrt{\frac{808549.36}{27} - 173^2} = 4.16 \text{ cm}\)