Question

SAT scores: Assume that in a given year the mean mathematics SAT score was 495, and the standard deviation was 111. A sample of 61 scores is chosen. Use the Th-84 Plus calculator. Part 1 of 5 (a) What is the probability that a randomly selected student scored above 550 on the mathematics SAT?

Answer 1

To find this probability, we can use the standard normal distribution and the Z-score formula:
`\( Z = \frac{{X - \mu}}{{\sigma}} \)`, where
`\( X \)` is the score,
`\( \mu \)` is the mean, and
`\( \sigma \)` is the standard deviation. For a score of 550, the Z-score is calculated as
`\( Z = \frac{{550 - 495}}{{111}} \approx 0.4955 \)`. Using a standard normal distribution table or a calculator, we find that the probability of Z being greater than 0.4955 is approximately 0.3523.

Explanation:

To determine the probability that a randomly selected student scored above 550 on the mathematics SAT, we employ the standard normal distribution. The process involves calculating the Z-score using the formula
`\( Z = \frac{{X - \mu}}{{\sigma}} \)`, where
`\( X \)` is the score,
`\( \mu \)` is the mean (495 in this case), and
`\( \sigma \)` is the standard deviation (111). For a score of 550, the Z-score is computed as
`\( Z = \frac{{550 - 495}}{{111}} \approx 0.4955 \)`.

Using statistical tables or a calculator, we determine the probability associated with this Z-score. In this context, we're interested in finding the area to the right of the Z-score on the standard normal distribution curve, as it represents scores above 550. The obtained probability is approximately 0.3523, indicating that there's a 35.23% chance that a randomly selected student scored above 550 on the mathematics SAT.

This process relies on the normalization of scores, allowing for a standardized comparison across different distributions. Understanding Z-scores and their corresponding probabilities is fundamental in interpreting the relative standing of individual scores within a given dataset.