There are two college entrance exams that are often taken by students, the ACT and the SAT. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1.The composite score on the SAT is approximately normally distributed with mean 1026and standard deviation 210.Suppose you scored 25on the ACT and 1179on the SAT. Which exam did you score better on? Justify your reasoning using the normal model.
To determine which exam you scored better on, we need to compare your scores relative to the mean and standard deviation of each exam
To determine which exam you scored better on, we need to compare your scores relative to the mean and standard deviation of each exam.
Let’s start with the ACT score of 25. We know that the mean for the ACT composite score is 21.1 with a standard deviation of 5.1. To use the normal model, we can calculate the z-score for this score using the formula:
z = (x – μ) / σ
where x is the given score, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:
z_ACT = (25 – 21.1) / 5.1 = 0.7647
The z-score tells us how many standard deviations your score is above or below the mean. In this case, a z-score of 0.7647 means your ACT score is approximately 0.7647 standard deviations above the mean.
Now let’s move on to the SAT score of 1179. The mean for the SAT composite score is 1026 with a standard deviation of 210. Applying the same formula, we can calculate the z-score:
z_SAT = (1179 – 1026) / 210 = 0.7286
Similarly, a z-score of 0.7286 for your SAT score indicates that it is approximately 0.7286 standard deviations above the mean.
Now, to compare the two scores, we need to compare the z-scores. Since both scores are positive, higher z-scores represent higher scores relative to the mean. Therefore, you performed better on the ACT compared to the SAT.
In summary, with an ACT score of 25, which corresponds to a z-score of 0.7647, and an SAT score of 1179, which corresponds to a z-score of 0.7286, it is evident that your ACT score is higher when compared to the mean and standard deviation of the ACT exam, thus implying that you scored better on the ACT compared to the SAT.
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