Final answer:
The standard deviation can be found using the formula sqrt((sigma((x - mu)²))/n), where sigma represents the sum of the squares of the differences between the values and the mean, mu is the mean, and n is the number of values. In this case, assuming a worst-case scenario, we can calculate the standard deviation.
Step-by-step explanation:
The standard deviation of a distribution can be found using the formula σ = sqrt((Σ(x - μ)²)/n), where σ is the standard deviation, Σ represents the sum of the squares of the differences between the values and the mean, μ is the mean, and n is the number of values. In this case, the mean is given as 72, and we know that at least 120 values fall within the interval 67-77. To find the standard deviation, we can assume worst-case scenario where all 40 remaining values fall outside the interval 67-77. This means we have 40 values outside the interval, and 120 values within the interval.
Now, let's calculate the sum of the squares of the differences between the values and the mean:
(40 * (67 - 72)²) + (120 * (72 - 72)²) + (40 * (77 - 72)²) = 40 + 0 + 40
Using this sum and the number of values, we can calculate the standard deviation:
σ = sqrt((40 + 0 + 40) / 160) = sqrt(80 / 160) = sqrt(0.5) ≈ 0.71