The caffeine content of 8-ounce cans of a certain cola drink is approximately normally distributed with mean 33 milligrams (mg). A randomly selected 8-ounce can containing 35 mg of caffeine is 1.2 standard deviations above the mean. Approximately what percent of 8-ounce cans of the cola have a caffeine content greater than 35 mg?
A. 196
B. 8%
C. 12%
D. 16%
E. 99%
Determine each of the following: a. 49 div 8. b. 49 mod 8. c. -105 mod 12
The Correct Answer and Explanation is:
Question 1: Normal Distribution and Z-Score Interpretation
We are given:
- Mean (μ) = 33 mg
- A sample value = 35 mg
- Z-score = 1.2
The Z-score represents how many standard deviations a value is from the mean. A Z-score of 1.2 means the value (35 mg) is 1.2 standard deviations above the mean.
To determine the percentage of cans with more than 35 mg, we want the area to the right of Z = 1.2 on the standard normal distribution curve.
Using a Z-table or calculator:
- P(Z < 1.2) ≈ 0.8849, meaning 88.49% of values fall below 35 mg.
- Therefore, P(Z > 1.2) = 1 – 0.8849 = 0.1151, or approximately 11.51%.
Rounded to the nearest whole number, the correct answer is:
C. 12%
Question 2: Modulo and Integer Division
a. 49 ÷ 8 (Integer Division):
We divide 49 by 8:
- 8 goes into 49 6 times (because 8 × 6 = 48, which is the largest multiple of 8 less than or equal to 49)
✅ Answer: 6
b. 49 mod 8:
This is the remainder when 49 is divided by 8.
- 49 ÷ 8 = 6 remainder 1
✅ Answer: 1
c. -105 mod 12:
Modulo with negative numbers follows this rule:
a mod b = the remainder r such that 0 ≤ r < b, and a ≡ r (mod b)
First, divide -105 by 12:
- -105 ÷ 12 = -9 (because 12 × -9 = -108, the closest lower multiple)
- The difference: -105 – (-108) = 3
✅ Answer: 3
This means:
- a. 49 ÷ 8 = 6
- b. 49 mod 8 = 1
- c. -105 mod 12 = 3
Summary
- The caffeine question involves reading the Z-score and finding the area under the normal curve. The answer is 12%, based on the percentage of data lying above a Z-score of 1.2.
- Integer division and modulo problems are straightforward applications of division with remainders, even for negative values (which must yield a non-negative remainder).