Final answer:
a. The probability of scoring higher than 70 is approximately 94.52%. b. The minimum score to earn an A grade is approximately 84.4. c. Approximately 74.86% of students have scores 5 or more points above the score that cuts off the lowest 25%.
Step-by-step explanation:
a. To find the probability that a person taking the examination scores higher than 70, we need to calculate the z-score and find the corresponding area under the normal curve.
The z-score is calculated as (x - mean) / standard deviation, where x is the score of interest. In this case, x = 70, mean = 78, and standard deviation = √(variance) = √(25) = 5.
Therefore, the z-score is (70 - 78) / 5 = -1.6. Using a standard normal distribution table or a calculator, we can find the area to the right of -1.6, which represents the probability of scoring higher than 70. The area, or probability, is 0.9452, or approximately 94.52%.
b. To find the minimum score a student must achieve to earn an A grade, we need to find the score that corresponds to the top 10% of the distribution. We can use the z-score formula to find the z-value that represents the top 10%. The z-value can be found using a standard normal distribution table or a calculator.
For the top 10%, the z-value is approximately 1.28. We can then use the z-score formula to find the corresponding score: x = mean + (z * standard deviation). Plugging in the values, we get x = 78 + (1.28 * 5) = 84.4. Therefore, the minimum score a student must achieve to earn an A grade is 84.4.
c. To approximate the proportion of students who have scores 5 or more points above the score that cuts off the lowest 25%, we first need to find the z-score that corresponds to the 25th percentile. The z-value can be found using a standard normal distribution table or a calculator.
For the 25th percentile, the z-value is approximately -0.674. We can then use the z-score formula to find the corresponding score: x = mean + (z * standard deviation). Plugging in the values, we get x = 78 + (-0.674 * 5) = 74.63. We need to find the proportion of students who have scores 5 or more points above this score.
We can find this by subtracting the area to the left of the score (25th percentile) from the total area under the curve, which represents the proportion of students who have scores above the score.
The area to the left of the score is approximately 0.2514, so the proportion of students who have scores 5 or more points above the score is approximately 1 - 0.2514 = 0.7486, or approximately 74.86%.