Final answer:
To determine the values between which 78% of a normal distribution with a mean of 8 and standard deviation of 3 lies, one would typically use a z-score table, since the Empirical Rule does not provide exact values for 78% of the data. We estimate the values will be slightly over one standard deviation from the mean, calculated as μ ± zσ.
Step-by-step explanation:
Given the normal distribution X with a mean (μ) of 8 and a standard deviation (σ) of 3, to find the two values between which 78% of the distribution lies, we can refer to the Empirical Rule.
However, the Empirical Rule specifically outlines percentages for 68%, 95%, and 99.7% of data within 1, 2, and 3 standard deviations from the mean respectively.
To find the range for 78%, which isn't directly outlined by the Empirical Rule, we would typically use a standard normal distribution table or a calculator to determine the z-scores that correspond to the cumulative areas of (1 - 0.78) / 2 = 0.11 (or the 11th percentile) and 1 - 0.11 = 0.89 (or the 89th percentile).
For the purpose of this question and because the Empirical rule does not provide exact values for 78%, we would estimate that 78% of values are slightly over one standard deviation from the mean.
This range can be calculated as μ ± zσ, where z is a z-score corresponding to 78% in a standard normal distribution table.