Question

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?

Answer 1

Complete question;

The distribution of lengths of salmon from a certain river is approximately normal with a standard deviation of 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

Option B is correct. The value that is closest to the mean of the distribution is 30inches.

The formula for calculating the z-score is expressed as:

$z=\frac{ x-\mu}{\sigma}$

Given the following parameters

$\sigma = 3.5\\x = 26in$

If 10 percent of salmon are longer than 30 inches, then:

• P(x>30) = 0.1

Using the z table to get the value corresponding to the mean area 0.1.

• The required z-score will be -1.285

Substitute the resulting parameters into the formula to get the mean of the distribution.

$z=\frac{x-\mu}{\sigma}\\-1.285=\frac{26-\mu}{3.5}\\-1.285 \times 3.5 = 26 - \mu\\ -4.4975 = 26 - \mu\\\mu = 26 + 4.4975\\\mu = 30.4975\\\mu \approx 30in$

Hence the value that is closest to the mean of the distribution is 30inches.

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