Answer :
The score separating the bottom 68% from the top 32% is approximately 44.4.
To find the score separating the bottom 68% from the top 32%, we need to determine the corresponding z-score and then convert it back to the original scale using the mean and standard deviation.
Since the normal distribution is symmetric, we know that the bottom 34% (half of the bottom 68%) lies below the mean, and the top 34% lies above the mean. To find the z-score that corresponds to the bottom 34%, we subtract this percentage from 0.5, resulting in 0.5 - 0.34 = 0.16.
Using a standard normal distribution table or a calculator, we can find the z-score corresponding to 0.16, which is approximately -0.994.
Next, we convert this z-score back to the original scale using the formula [tex]z = (x - μ) / σ[/tex], where z is the z-score, x is the value we want to find, μ is the mean, and σ is the standard deviation. Rearranging the formula, we have[tex]x = z * σ + μ.[/tex]
Plugging in the values, we get [tex]x = -0.994 * 6.5 + 37.9[/tex], which gives us approximately 44.4.
In summary, the score separating the bottom 68% from the top 32% is approximately 44.4.
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