(i) To find the mean \(\bar{x}\), sum the times and divide by the number of students:

\(\bar{x} = \frac{15 + 10 + 48 + 10 + 19 + 14 + 16}{7} = \frac{132}{7} = 18.9\)

To find the standard deviation \(sd\), use the formula:

\(sd = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}\)

Calculate each squared deviation:

• \((15 - 18.9)^2 = 15.21\)
• \((10 - 18.9)^2 = 79.21\)
• \((48 - 18.9)^2 = 841.21\)
• \((10 - 18.9)^2 = 79.21\)
• \((19 - 18.9)^2 = 0.01\)
• \((14 - 18.9)^2 = 24.01\)
• \((16 - 18.9)^2 = 8.41\)

Sum of squared deviations: \(15.21 + 79.21 + 841.21 + 79.21 + 0.01 + 24.01 + 8.41 = 1057.27\)

\(sd = \sqrt{\frac{1057.27}{7}} = \sqrt{151.04} = 12.3\)

(ii) The median is the most appropriate measure of central tendency because it is not affected by the outlier (48).

(iii) The mode is inappropriate because it is 10, which is the lowest value and not representative of the data. The mean is inappropriate because it is affected by the outlier (48), which skews the average.

(i) Since the standard deviation is $0, all rides must cost the same, which is the mean of $2.50. Therefore, the roller coaster and the water slide each cost $2.50 per ride.

(ii) Let the cost of the revolving drum be $x. The equation for the mean cost is:

\(1 \times 2.5 + 3 \times 2.5 + 6 \times x = 3.76 \times 10\)

\(6x = 37.6 - 10\)

\(x = 4.6\)

\(The cost of the revolving drum is $4.60. Now, calculate the variance:\)

\(\sigma^2 = \frac{(2.5^2 \times 1 + 2.5^2 \times 3 + 4.6^2 \times 6)}{10} - 3.76^2\)

\(\sigma = 1.03\)

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

\(\bar{x} = \frac{32 + 35 + 45 + 37 + 38 + 44 + 33 + 39 + 36 + 45}{10} = \frac{384}{10} = 38.4 \text{ mm}\)

\(To find the standard deviation, first calculate the variance:\)

\(s^2 = \frac{1}{10} \left( (32 - 38.4)^2 + (35 - 38.4)^2 + (45 - 38.4)^2 + (37 - 38.4)^2 + (38 - 38.4)^2 + (44 - 38.4)^2 + (33 - 38.4)^2 + (39 - 38.4)^2 + (36 - 38.4)^2 + (45 - 38.4)^2 \right)\)

\(s^2 = \frac{1}{10} (40.96 + 11.56 + 43.56 + 1.96 + 0.16 + 31.36 + 29.16 + 0.36 + 5.76 + 43.56)\)

\(s^2 = \frac{1}{10} (208.4) = 20.84\)

Then, the standard deviation is:

\(s = \sqrt{20.84} \approx 4.57 \text{ mm}\)

(i) In the first round, 32 teams play, resulting in 16 matches. Therefore, 16 teams lose and play only 1 match.

(ii) In the second round, 16 teams play, resulting in 8 matches. Therefore, 8 teams lose and play exactly 2 matches.

(iii) The frequency table is constructed as follows:

(iv) To calculate the mean and variance:
Total matches played = 16(1) + 8(2) + 4(3) + 2(4) + 2(5) = 62
Mean = \(\frac{62}{32} = 1.9375 \approx 1.94\)
For variance, calculate \(\sum f x^2 = 16(1^2) + 8(2^2) + 4(3^2) + 2(4^2) + 2(5^2) = 166\)
Variance = \(\frac{166}{32} - \left(\frac{62}{32}\right)^2 = 1.43\)

To find the median, first arrange the data in ascending order: 40, 40, 41, 42, 45, 47, 48, 49, 50, 51, 352.

The median is the middle value, which is the 6th value in this ordered list: 47.

The median is $47,000.

The mean is not suitable because the data contains an outlier (352), which skews the mean. The mode is not suitable because it does not represent the central tendency effectively in this context.