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What percent of scores fall between 70 and 80 on a test with a normal distribution, mean 60, and standard deviation 15?

Answer :

Final answer:

The percent of scores falling between 70 and 80 on a normally distributed test with a mean of 60 and a standard deviation of 15 is approximately 15.96%.

Explanation:

To find what percent of scores fall between 70 and 80 on a test with a normal distribution, mean of 60, and standard deviation of 15, we need to calculate the Z-scores for both the lower bound (70) and the upper bound (80) and then use the standard normal distribution table to find the corresponding probabilities.

The formula for the Z-score is:

Z = (X - μ) / σ

Where X is the score, μ is the mean, and σ is the standard deviation.

For the lower bound (70):
Z = (70 - 60) / 15 = 10 / 15 ≈ 0.67

For the upper bound (80):
Z = (80 - 60) / 15 = 20 / 15 ≈ 1.33

We then use a Z-table or calculator to find the percent of scores falling between these two Z-scores. The area under the curve between these two values represents the percentage of scores that fall between 70 and 80.

For Z = 0.67, the table gives us the area to the left, which is approximately 0.7486. For Z = 1.33, the area to the left is approximately 0.9082.

To find the percentage between the two Z-scores, we subtract the lower Z-score's area from the higher Z-score's area:

Percentage = 0.9082 - 0.7486 = 0.1596

Therefore, approximately 15.96% of the scores fall between 70 and 80.

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