Suppose that you got a score of X=78 on an English test for which the mean was (mu)=70 and the standard deviation was (lower case sigma)=10. Also, suppose that you got a scores of X=64 on a Spanish test with mean(mu)=50 and (lower case sigma)=7. For which test would you expect the better grade? Explain your answer.
(this is for a population)
On the English test you were 0.8-sigma above the average, and on the Spanish test you were also 2.0 sigma above the average. The Spanish test result deserves the better grade.
To determine which test you would expect the better grade, we need to compare the individual scores to their respective means and standard deviations.
For the English test:
1. Calculate the z-score for the English test score using the formula: z = (X - mu) / sigma, where X is the individual score, mu is the mean, and sigma is the standard deviation.
z = (78 - 70) / 10
z = 0.8
2. Look up the z-score in the standard normal distribution table or use a calculator to find the corresponding percentile. In this case, the z-score of 0.8 corresponds to a percentile of approximately 79.32%.
For the Spanish test:
1. Calculate the z-score for the Spanish test score using the same formula: z = (X - mu) / sigma.
z = (64 - 50) / 7
z = 2
2. Look up the z-score in the standard normal distribution table or use a calculator to find the corresponding percentile. In this case, the z-score of 2 corresponds to a percentile of approximately 97.72%.
Comparing the percentiles from both tests, we can see that the Spanish test has a higher percentile, indicating a better relative performance compared to the English test. Therefore, we would expect a better grade on the Spanish test.