In a university, the scores of a math final exam are normally distributed with a mean of 70 and a standard deviation of 10. What is the probability that a randomly selected student scored more than 85?
Answer
Step 2: The Z-score of 1.5 indicates that the score of 85 is 1.5 standard deviations above the mean. Now, we need to find the area under the curve to the right of Z = 1.5, i.e., the probability that a randomly chosen score is greater than 85. This is given by 1 - probability of Z upto 1.5. The probability of Z upto 1.5 can be looked up in the standard normal distribution table or calculated using statistical software, and it is approximately 0.9332. So, the probability that Z > 1.5 is given by \(1 - 0.9332 = 0.0668\).
Step 1 ：Step 1: Standardize the score by subtracting the mean from it and then dividing by the standard deviation. This gives us a Z-score, which represents how many standard deviations away from the mean our score is. This calculation is done as follows: \(Z = \frac{X - \mu}{\sigma}\) where X is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Substituting the given values, we get \(Z = \frac{85 - 70}{10} = 1.5\).
Step 2 ：Step 2: The Z-score of 1.5 indicates that the score of 85 is 1.5 standard deviations above the mean. Now, we need to find the area under the curve to the right of Z = 1.5, i.e., the probability that a randomly chosen score is greater than 85. This is given by 1 - probability of Z upto 1.5. The probability of Z upto 1.5 can be looked up in the standard normal distribution table or calculated using statistical software, and it is approximately 0.9332. So, the probability that Z > 1.5 is given by \(1 - 0.9332 = 0.0668\).
