Answer:
(a) To find the sample mean x, we need to calculate the sum of the products of each value of X and its corresponding frequency, and then divide it by the total number of observations.
Sum of (X * Frequency) = (1 * 3) + (2 * 4) + (3 * 7) + (4 * 12) + (5 * 12) + (6 * 2) = 3 + 8 + 21 + 48 + 60 + 12 = 152
Total number of observations = 3 + 4 + 7 + 12 + 12 + 2 = 40
Sample mean x = Sum of (X * Frequency) / Total number of observations = 152 / 40 = 3.80 (rounded to two decimal places)
Therefore, the sample mean x is 3.80.
(b) To find the sample standard deviation s, we first need to calculate the squared deviations from the mean for each value of X, multiply them by their corresponding frequencies, and then sum them up. After that, we divide by the total number of observations and take the square root.
Squared deviations from the mean = [(1 - 3.80)^2 * 3] + [(2 - 3.80)^2 * 4] + [(3 - 3.80)^2 * 7] + [(4 - 3.80)^2 * 12] + [(5 - 3.80)^2 * 12] + [(6 - 3.80)^2 * 2]
= [(-2.80)^2 * 3] + [(-1.80)^2 * 4] + [(-0.80)^2 * 7] + [(0.20)^2 * 12] + [(1.20)^2 * 12] + [(2.20)^2 * 2]
= [7.84 * 3] + [3.24 * 4] + [0.64 * 7] + [0.04 * 12] + [1.44 * 12] + [4.84 * 2]
= 23.52 + 12.96 + 4.48 + 0.48 + 17.28 + 9.68
= 68.40
Sample variance = Sum of (Squared deviations from the mean) / Total number of observations = 68.40 / 40 = 1.71
Sample standard deviation s = Square root of Sample variance = √1.71 ≈ 1.31 (rounded to two decimal places)
Therefore, the sample standard deviation s is approximately 1.31.
(c) The histogram of the data would have a horizontal axis labeled "Number of Sneakers" and a vertical axis labeled "Frequency." The histogram would have 6 bars representing the values 1, 2, 3, 4, 5, and 6. The heights of the bars would correspond to their respective frequencies: 3, 4, 7, 12, 12, and 2.
