Question

(1 point)
mike took 4 courses last semester: spanish, history, biology, and calculus. the means and standard deviations for the final exams, and mike’s scores are given in the table below. convert mike’s score into z scores. enter z - scores to 2 decimal places.

subject	mean	stand. dev.	mike’s score	mike’s z score
history	53	16	61
biology	77	10	89.5
calculus	70	12	67

on what exam did mike have the highest relative score?
(enter the subject.)
note: you can earn partial credit on this problem
you are in the reduced scoring period: all additional work done counts 85% of the original.

Explanation:

The formula for calculating the z - score is \( z=\frac{x - \mu}{\sigma} \), where \( x \) is the individual's score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. We will calculate the z - score for each of Mike's courses.

Step 1: Calculate z - score for Spanish

For Spanish, \( x = 35 \), \( \mu=44 \), \( \sigma = 12 \)
\( z=\frac{35 - 44}{12}=\frac{- 9}{12}=- 0.75 \)

Step 2: Calculate z - score for History

For History, \( x = 61 \), \( \mu = 53 \), \( \sigma=16 \)
\( z=\frac{61 - 53}{16}=\frac{8}{16} = 0.50 \)

Step 3: Calculate z - score for Biology

For Biology, \( x=89.5 \), \( \mu = 77 \), \( \sigma = 10 \)
\( z=\frac{89.5-77}{10}=\frac{12.5}{10}=1.25 \)

Step 4: Calculate z - score for Calculus

For Calculus, \( x = 67 \), \( \mu=70 \), \( \sigma = 12 \)
\( z=\frac{67 - 70}{12}=\frac{-3}{12}=- 0.25 \)

Now we compare the z - scores: Spanish: - 0.75, History: 0.50, Biology: 1.25, Calculus: - 0.25. The highest z - score is 1.25 which corresponds to the Biology exam.

Answer:

The z - scores are: Spanish: \(-0.75\), History: \(0.50\), Biology: \(1.25\), Calculus: \(-0.25\)

The exam on which Mike had the highest relative score is Biology.