Question

1. a professor x gave a factoring exam (out of 100) and the class results were summarized by μ = 78.3 and σ=3.5

(a) if a student scored 86, would this be usual or unusual? why? \t\t\tusual/unusual (choose one)
\t\t\t\t\t\t\t\t\t\twhy? ___________________________

(b) if a student scored 72, would this be usual or unusual? why? \t\t\tusual/unusual (choose one)
\t\t\t\t\t\t\t\t\t\twhy? ___________________________

1. another professor y gave a factoring exam (out of 100 as well) and the class results were summarized

by μ = 81.1 and σ = 4.1. at least what percent of students scored between 62.65 and 99.55?
_________________________________

Explanation:

Step1: Calculate z-score for 86

$z = \frac{86 - 78.3}{3.5} \approx 2.2$

Step2: Determine usual/unusual

2.2 > 2, so unusual.

Step1: Calculate z-score for 72

$z = \frac{72 - 78.3}{3.5} = -1.8$

Step2: Determine usual/unusual

-1.8 within [-2,2], so usual.

Step1: Find k for bounds

$k = \frac{81.1 - 62.65}{4.1} = 4.5$

Step2: Apply Chebyshev's theorem

$1 - \frac{1}{4.5^2} \approx 95.1\%$

Answer:

Unusual, because its z-score ≈2.2 > 2.
Usual, because its z-score = -1.8 within [-2,2].
At least approximately 95.1% of students scored between 62.65 and 99.55.