Answer:
The closest to the mean of the distribution is 30 inches ⇒ B
Step-by-step explanation:
The formula of z-score is z = (x - μ)/σ , where x is the score, μ is the mean, and σ is the standard deviation
∵ The lengths of salmon from a certain river is approximately
normal with standard deviation 3.5 inches
∴ σ = 3.5 inches
∵ 10 percent of salmon are longer than 30 inches
- That means P(x > 30) = 10%
∵ 10% = [tex]\frac{10}{100}[/tex] = 0.10
∴ P(x > 30) = 0.10
Use the normal distribution table to find the z-score that corresponds to 0.10 of the distribution's area
∵ x = 26 inches
- The value of x is smaller than 30, so z-score is -ve
∵ z-score of corresponding area 0.10 is -1.285
- Substitute it in the formula of z-score above
∴ -1.285 = (26 - μ)/3.5
- Multiply both sides by 3.5
∴ -4.4975 = 26 - μ
- Add μ to both sides
∴ μ - 4.4975 = 26
- Add 4.4975 to both sides
∴ μ = 30.4975
- Round it to the nearest whole number
∴ μ ≅ 30
∴ The closest to the mean of the distribution is 30 inches
