Final answer:
To calculate z-scores for given vehicle speeds, first find the mean and standard deviation of the set. Then, use the z-score formula for each data point: A. 74 Z = 0.43, B. 65 Z = -0.54, C. 58 Z = -1.29.
Step-by-step explanation:
The question involves calculating the z-scores for given data values, which is a concept in statistics. To calculate the z-score for a data value, you must first determine the mean (average) and the standard deviation of the data set. The z-score is then found using the formula: z = (X - μ) / σ, where X is the data value, μ (mu) is the mean, and σ (sigma) is the standard deviation. Now, let's find the mean and the standard deviation for the speeds of the vehicles:
The mean speed: (Σ50 + 74 + 65 + 58 + 71 + 65 + 61 + 68 + 55 + 72 + 81 + 60) / 12 = 70.
The sum of the squared differences from the mean: (Σ(X - μ)^2) = (20^2 + 4^2 + (-5)^2 + (-12)^2 + 1^2 + (-5)^2 + (-9)^2 + (-2)^2 + (-15)^2 + 2^2 + 11^2 + (-10)^2).
Calculating the sum gives us 946, and the standard deviation is the square root of (total sum of squares / number of values - 1), which is √(946 / 11) = √86 approximately equals to 9.27.
We can now calculate the z-scores:
-
- For 74: z = (74 - 70) / 9.27 ≈ 0.43
-
- For 65: z = (65 - 70) / 9.27 ≈ -0.54
-
- For 58: z = (58 - 70) / 9.27 ≈ -1.29
The final answers for the z-scores are:
-
- A. 74 Z = 0.43
-
- B. 65 Z = -0.54
-
- C. 58 Z = -1.29