for each of the following, identify the exam score that should lead to the better grade. In each case, explain your answer.
A score of x=56, on an exam with the mean =50 and the standard deviation of 4; or a score of x=60 on an exam with the mean=50 and the standard deviation =20
Z = (score-mean)/SD
Which test has the highest Z score?
To determine which exam score should lead to the better grade, we need to compare the scores relative to their respective distributions.
For the first case, where x=56, on an exam with a mean of 50 and a standard deviation of 4, we need to calculate how many standard deviations above the mean this score falls.
First, we calculate the z-score using the formula: z = (x - μ) / σ, where z is the z-score, x is the exam score, μ is the mean, and σ is the standard deviation.
Plugging in the values:
z = (56-50) / 4
z = 6 / 4
z = 1.5
The z-score of 1.5 tells us that the score of 56 is 1.5 standard deviations above the mean.
For the second case, where x=60, on an exam with a mean of 50 and a standard deviation of 20, we use the same process to calculate the z-score:
z = (60-50) / 20
z = 10 / 20
z = 0.5
The z-score of 0.5 indicates that the score of 60 is 0.5 standard deviations above the mean.
Now, let's consider the implications of these z-scores on the grades. Typically, a higher z-score results in a better grade, as it indicates a higher performance relative to the rest of the class.
In this comparison, a z-score of 1.5 (for x=56) is higher than 0.5 (for x=60). Therefore, the score of 56 on the first exam should lead to a better grade compared to the score of 60 on the second exam.
To determine which exam score will lead to the better grade, we need to understand the concept of z-scores and how they relate to the mean and standard deviation of a distribution.
The z-score indicates how many standard deviations an individual score is above or below the mean of a distribution. It is calculated by subtracting the mean from the individual score and then dividing the difference by the standard deviation.
Let's calculate the z-score for both scores:
For a score of x=56, mean=50, and standard deviation=4:
z-score = (56 - 50) / 4 = 6 / 4 = 1.5
For a score of x=60, mean=50, and standard deviation=20:
z-score = (60 - 50) / 20 = 10 / 20 = 0.5
Now, comparing the z-scores, we can see that a score of 56 has a z-score of 1.5, while a score of 60 has a z-score of 0.5. This implies that a score of 56 is further above the mean of its distribution than a score of 60 is above the mean of its distribution.
Since a z-score higher than the mean indicates a better performance relative to other students, a score of 56, with a higher z-score of 1.5, should lead to a better grade than a score of 60 with a z-score of 0.5.
It's important to note that the grading scale and criteria may vary depending on the specific context, so this explanation is based solely on the comparison of z-scores in relation to the mean and standard deviation.