Question

Scores on an exam were normally distributed. Ten percent of the scores were below 64 and 80% were below 81 . Find the mean and standard deviation of the scores Mean = Standard deviation =

Answer 1

To find the mean and standard deviation from the given percentiles of a normal distribution, set up equations using the z-scores for the corresponding percentiles (-1.28 for 10% and +0.84 for 80%). Solving these equations simultaneously yields a mean of 76.5 and a standard deviation of 11.5.

Step-by-step explanation:

Given that the scores on an exam were normally distributed, it's possible to use the given percentiles to find the mean and standard deviation of the scores. The fact that 10% of scores are below 64 and 80% below 81 matches certain z-scores in the standard normal distribution.

First, the z-score corresponding to the 10th percentile (which is -1.28) gets lined up with 64:

z = (X - μ) / σ
-1.28 = (64 - μ) / σ

Next, the z-score corresponding to the 80th percentile (which is +0.84) aligns with 81:

+0.84 = (81 - μ) / σ

These two equations can be solved simultaneously to find μ (the mean) and σ (the standard deviation). After solving, we find:

μ = mean = 76.5

σ = standard deviation = 11.5

Answer 2

To find the mean and standard deviation, we use the z-score formula and the given information. After Simplifying we get σ = 17.52 , μ = 87.26.

Step-by-step explanation:

To find the mean and standard deviation of the scores, we can use the z-score formula and the information given. The z-score formula is z = (x - μ) / σ, where z is the z-score, x is the raw score, μ is the mean, and σ is the standard deviation.

Given that 10% of scores were below 64, we can find the z-score for 64 using the formula: (64 - μ) / σ = -1.28. Solving for μ, we get μ = 64 + 1.28σ.

Similarly, we can find another equation using the fact that 80% of the scores were below 81: (81 - μ) / σ = 0.84. Solving for μ, we get μ = 81 - 0.84σ.

Setting these two equations equal to each other, we can solve for σ: 64 + 1.28σ = 81 - 0.84σ. Simplifying and isolating σ, we get σ = 17.52.

Substituting this value of σ back into one of the original equations, we can solve for μ: μ = 64 + 1.28(17.52) = 87.26.