Final answer:
The z-score for a value of 386 in a normally distributed dataset with mean 331 and standard deviation 43 is approximately 1.2791. Z-scores indicate how many standard deviations a value is from the mean, with positive values above the mean. Reference to a Z-table allows one to determine the probability associated with specific z-scores.
Step-by-step explanation:
Finding Z-Scores from Given Values
To find the z-score for a value in a normally distributed set of data, you use the formula z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. In the case of a value of 386 with a mean (average) of 331 and standard deviation of 43, the z-score would be (386 - 331) / 43 = 55 / 43 ≈ 1.2791. This z-score indicates that the value of 386 is approximately 1.2791 standard deviations above the mean.
Interpreting Z-Scores
Z-scores provide understanding of how far a specific data point is from the mean in terms of the number of standard deviations. A z-score of 0 indicates the value is exactly at the mean. Positive z-scores indicate values above the mean, and negative z-scores indicate values below the mean. For example, a z-score of -0.40 for x₁ = 325 means it is 0.40 standard deviations below the mean, while a z-score of 1.5 for x₂ = 366.21 indicates it is 1.5 standard deviations above the mean.
Using Z-scores with Z-table
To interpret probabilities using z-scores, you reference a Z-table which provides the area under the normal curve to the left of a given z-score. This area corresponds to the probability of a value being less than or equal to that value. For instance, with a z-score of -0.40, the area to the left is 0.3446, indicating a 34.46% probability of a randomly selected data point being less than or equal to x₁ = 325.