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.