The final exam scores in a statistics class were normally distributed with a mean of 63 and a standard deviation of five. Find the probability that a randomly selected student scored more than 68 on the exam. (3 marks)
Question
The final exam scores in a statistics class were normally distributed with a mean of 63 and a standard deviation of five. Find the probability that a randomly selected student scored more than 68 on the exam. (3 marks)
Solution
To solve this problem, we need to use the concept of Z-scores in statistics.
Step 1: Calculate the Z-score. The Z-score is a measure of how many standard deviations an element is from the mean. The formula for the Z-score is:
Z = (X - μ) / σ
where: X is the score we are interested in, which is 68. μ is the mean, which is 63. σ is the standard deviation, which is 5.
So, Z = (68 - 63) / 5 = 1.
Step 2: Look up the Z-score in the Z-table. The Z-table tells us what percentage of the population is below a certain Z-score. For a Z-score of 1, the value is approximately 0.8413. This means that about 84.13% of the scores are below 68.
Step 3: Since we are interested in the scores above 68, we need to subtract the value we found from 1 (because the total probability under the normal curve is 1). So, 1 - 0.8413 = 0.1587.
Therefore, the probability that a randomly selected student scored more than 68 on the exam is approximately 0.1587, or 15.87%.
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