Problem

Suppose that the scores on a reading ability test are normally distributed with a mean of 70 and a standard deviation of 10 . What proportion of individuals score more than 68 points on this test? Round your answer to at least four decimal places.

Answer

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Answer

Final Answer: The proportion of individuals who score more than 68 points on the test is approximately \(\boxed{0.5793}\).

Steps

Step 1 :The problem is asking for the proportion of individuals who score more than 68 points on the test. This is a problem of finding the area under the curve of a normal distribution. The area under the curve to the right of a given point in a normal distribution represents the proportion of individuals who score above that point.

Step 2 :To solve this problem, we need to convert the score of 68 to a z-score, which represents how many standard deviations the score is away from the mean. The formula for calculating the z-score is \((X - μ) / σ\), where X is the score, μ is the mean, and σ is the standard deviation.

Step 3 :Substituting the given values into the formula, we get \((68 - 70) / 10 = -0.2\). So, the z-score is -0.2.

Step 4 :After calculating the z-score, we can use the standard normal distribution table to find the proportion of individuals who score above that z-score. The proportion corresponding to a z-score of -0.2 is approximately 0.5793.

Step 5 :This means that about 57.93% of individuals score more than 68 points on the test.

Step 6 :Final Answer: The proportion of individuals who score more than 68 points on the test is approximately \(\boxed{0.5793}\).

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