Question

The scores on a national achievement test were approximately normally distributed, with a mean of 540 and a standard deviation of \(110 .\) a. If you achieved a score of 680 , how far, in standard deviations, did your score depart from the mean? b. What percentage of those who took the examination scored higher than you?

Short answer

a) The score of 680 is approximately 1.27 standard deviations above the mean. b) Approximately 10.20% of the students who took the examination scored higher than 680.

Step-by-step solution

Calculate the z-score for 680

The z-score formula is as follows: \(z = \frac{X - \mu}{\sigma}\) Where: \(z\) = z-score \(X\) = raw score \(\mu\) = mean \(\sigma\) = standard deviation For our problem: \(X = 680\) \(\mu = 540\) \(\sigma = 110\) Now, we can plug these values into the z-score formula: \(z = \frac{680 - 540}{110}\)

Calculate the z-score value

Now we can calculate the z-score: \(z = \frac{140}{110} \approx 1.27\) This means that the score of 680 is approximately 1.27 standard deviations above the mean.

Find the percentage of students who scored higher

We must find the proportion of the area to the right of the z-score in a normal distribution. By using a z-table, we look for the area corresponding to a z-score of 1.27. We find that the area to the left of the z-score is approximately 0.8980. Since we are interested in the percentage of students who scored higher (to the right of z-score), we calculate the area to the right by subtracting the left area from 1: \(1 - 0.8980 = 0.1020\) Now, we convert this proportion to a percentage by multiplying it by 100: \(0.1020 * 100 = 10.20 \%\) Hence, approximately \(10.20 \%\) of the students who took the examination scored higher than 680.