Final answer:
To calculate the probability of various scores on a mathematics assessment test, you can use z-scores and a standard normal distribution table or calculator. For each part of the question, we can calculate the z-scores and find the corresponding probabilities or percentiles. The lowest score to place a student in the top 10% is approximately 685.18, and the highest score to place a student in the bottom 15% is approximately 415.39.
Step-by-step explanation:
a) Probability of score less than 600:
To calculate the probability, we need to convert the score to a z-score using the formula z = (x - mean) / standard deviation. For a score of 600, the z-score would be (600 - 534) / 112 = 0.5893. Using a z-table or calculator, we can find that the probability is approximately 0.7224, or 72.24%.
b) Probability of score between 515 and 625:
First, we calculate the z-scores for both scores: (515 - 534) / 112 = -0.1696 and (625 - 534) / 112 = 0.8125. Using a z-table or calculator, we can find the probabilities for both z-scores. The probability of a score below 515 is approximately 0.4345, and the probability of a score below 625 is approximately 0.7910. The probability between 515 and 625 is the difference of these probabilities, which is approximately 0.7910 - 0.4345 = 0.3565, or 35.65%.
c) Percentage of students with score > 540:
To calculate the percentage, we need to convert the score to a z-score: (540 - 534) / 112 = 0.0536. Using a z-table or calculator, we can find the probability of a score below 540, which is approximately 0.5200. The percentage of students with a score greater than 540 is 100% - 52% = 48%.
d) Lowest score in the top 10%:
To find the lowest score in the top 10%, we need to find the z-score that corresponds to the 90th percentile. Using a z-table or calculator, we can find this z-score to be approximately 1.282. We can then use the z-score formula z = (x - mean) / standard deviation to find the corresponding score: 1.282 = (x - 534) / 112. Solving for x, we get x = 1.282 * 112 + 534 = 685.18. So the lowest score to be in the top 10% is approximately 685.18.
e) Highest score in the bottom 15%:
Similar to the previous question, we need to find the z-score that corresponds to the 15th percentile. Using a z-table or calculator, we can find this z-score to be approximately -1.036. Using the z-score formula, we can find the corresponding score: -1.036 = (x - 534) / 112. Solving for x, we get x = -1.036 * 112 + 534 = 415.39. So the highest score to be in the bottom 15% is approximately 415.39.