The distribution of lengths of salmon from a certain river is approximately normal with standard deviation 3.5 inches. If 10 percent of salmon are longer than 30 inches, which of the following is closest to the mean of the distribution? 26 inches A 28 inches B 30 inches C 33 inches D 34 inches

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% = = 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