Question

The grades of an exam whose mean is 82 and whose standard deviation is 14 are normally distributed. Anyone who scores below 55 will be retested. What percentage does this represent

Answer 1

Approximately 2.7% of students are expected to score below 55 on an exam with a mean of 82 and a standard deviation of 14 when grades are normally distributed.

Step-by-step explanation:

The original question asks what percentage of students are expected to score below 55 on an exam if the grades are normally distributed with a mean of 82 and a standard deviation of 14. To solve this, we need to calculate the z-score for a grade of 55, which tells us how many standard deviations away from the mean this score is. The formula for a z-score is given by:

Z = (X - μ) / σ

where Z is the z-score, X is the value in question, μ is the mean, and σ is the standard deviation. Plugging in the values we get:

Z = (55 - 82) / 14 = -27 / 14 ≈ -1.93

Next, we use a z-score table or calculator to find the percentage of students scoring below this z-score. A z-score of -1.93 corresponds to roughly 2.7% of the data lying below it in a normal distribution. Thus, approximately 2.7% of students would be expected to score below 55 and be retested.

Answer 2

This represents 2.6803%

Step-by-step explanation:

We start by calculating the z-score

Mathematically;

z-score = (x - mean)/SD

here our x is 55

mean = 82

SD = 14

z-score = (55-82)/14

z-score = -1.93

So the probability that a test taker will be retested will be;

P( x < -1.93)

We use the standard normal distribution table for this

P( x < -1.93) = 0.026803

If we convert this to percentage, we have 2.6803%