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The scores on a standardized test given to third-graders are normally distributed with a mean of 150 and a standard deviation of 30. Which test score closest to the 45th percentile? Round to the nearest whole number.

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Final answer:

To find the test score closest to the 45th percentile for the third-graders' scores, you use the z-score for the 45th percentile, which is approximately -0.125. Then apply the conversion formula: Actual Score = Mean + (Z-score * Standard Deviation), which gives 146 as the closest test score to the 45th percentile after rounding to the nearest whole number.

Step-by-step explanation:

The question asks to determine the test score closest to the 45th percentile of a normally distributed set of third-grader scores with a mean of 150 and a standard deviation of 30. To find the score closest to the 45th percentile, we can use the normal distribution and z-scores. The z-score correlates to the number of standard deviations a value is from the mean.

To find the z-score for the 45th percentile, we would use the inverse normal function, often denoted as invNorm. However, a common approximation for the 50th percentile is a z-score of 0 because it is the mean. Since the 45th percentile is close to the 50th percentile, we expect the z-score to be close to 0, but slightly negative because it is below the mean.

Using a standard z-table, or the invNorm function on a calculator, we find that the z-score for the 45th percentile is approximately -0.125. Now we apply the formula to convert the z-score to the actual score:

Actual Score = Mean + (Z-score * Standard Deviation)

Putting in the values, we have:

Actual Score = 150 + (-0.125 * 30)

Calculating the actual score:

Actual Score = 150 - 3.75

Actual Score = 146.25

After rounding to the nearest whole number, we get 146 as the test score closest to the 45th percentile.

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