he following table lists the probability distribution of the number of shoplifting incidents that occur during a day at a certain shopping center:
Number of shoplifting incidents 0 1 2 3 4
Probability 0.1 0.2 0.25 0.3 0.15
Determine the standard deviation for the number of shoplifting incidents during a given day.
1.43
2.51
1.1
2.02
1.21
To calculate the standard deviation, we first need to find the mean of the distribution.
Mean (μ) = Σ(x * P(x)), where x represents the number of shoplifting incidents and P(x) is the probability of that number of incidents occurring.
Mean = (0 * 0.1) + (1 * 0.2) + (2 * 0.25) + (3 * 0.3) + (4 * 0.15)
Mean = 0 + 0.2 + 0.5 + 0.9 + 0.6
Mean = 2.2
Next, we calculate the variance using the formula:
Variance (σ^2) = Σ((x - μ)^2 * P(x))
Variance = ((0-2.2)^2 * 0.1) + ((1-2.2)^2 * 0.2) + ((2-2.2)^2 * 0.25) + ((3-2.2)^2 * 0.3) + ((4-2.2)^2 * 0.15)
Variance = (2.2^2 * 0.1) + (1.2^2 * 0.2) + (0.2^2 * 0.25) + (0.8^2 * 0.3) + (1.8^2 * 0.15)
Variance = (4.84 * 0.1) + (1.44 * 0.2) + (0.04 * 0.25) + (0.64 * 0.3) + (3.24 * 0.15)
Variance = 0.484 + 0.288 + 0.01 + 0.192 + 0.486
Variance = 1.44
Finally, the standard deviation is the square root of the variance:
Standard Deviation = √1.44
Standard Deviation = 1.2
Therefore, the standard deviation for the number of shoplifting incidents during a given day is 1.2.
The closest option provided is 1.21, which is the correct answer.