Final answer:
The number of cases with scores above 18 in a sample of 1000 cases, with a normal distribution and given mean and standard deviation, is estimated using z-scores. The z-score for a score of 18 is 1.6. Therefore, it's estimated that between 100 and 200 scores are above 18.
Step-by-step explanation:
The question asks for the number of scores above 18 in a sample of 1000 cases, given a normal distribution with a mean of 14 and a standard deviation of 2.5. To find the answer, we use the concept of z-scores, which measure how many standard deviations an element is from the mean. The z-score for a score of 18 is calculated as follows:
Z = (X - μ) / σ
Where X is the score, μ is the mean, and σ is the standard deviation. So, for a score of 18:
Z = (18 - 14) / 2.5 = 4 / 2.5 = 1.6
Next, we find the proportion of scores that lie above a z-score of 1.6 using standard normal distribution tables or a calculator. This gives us the probability that a single score is above 18. To find the number of scores above 18 out of 1000, we multiply this probability by 1000.
Without the exact probability value, we cannot do the calculation, but we know the probability corresponding to a z-score of 1.6 will be less than 0.5 (since 0 is the mean of the z-score distribution), and thus, less than 50% of the sample will be above 18. Therefore, the number of cases above 18 will be less than 500. From the given options, the most accurate estimate is between 100 and 200 scores, option 2.