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The speeds of random vehicles along a stretch of highway is recorded. {50, 74, 65, 58, 71, 65, 61, 68, 55, 72, 81, 60} Find the z-scores for each of the following data values.

A. 74 Z =

B. 65 Z =

C. 58 Z =

1 Answer

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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:




  1. For 74: z = (74 - 70) / 9.27 ≈ 0.43

  2. For 65: z = (65 - 70) / 9.27 ≈ -0.54

  3. 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

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