Smolts are young salmon at a stage when their skin becomes covered with silvery

Question

Smolts are young salmon at a stage when their skin becomes covered with silvery scales and they start to migrate from freshwater for migration. The distribution of relectances is approximately normal with a mean of 50.89 and an SD OF 8.8. show work. a) one smelt has a reflectance of 69.2. What percentile does this place him in? b) Another smolts has a reflectance of 42.8. What percentile does this place him in? c) If only the smolts with the highest 5% reflectance are considered prize catches. What reflectance level must a smelt have to be considered a prize catch? d) What reflectance is necessary to be at the third quartile?

Explanation / Answer

a) P(X < 69.2)

= P((X - mean)/sd < (69.2 - mean)/sd)

= P(Z < (69.2 - 50.89)/8.8)

= P(Z < 2.08)

= 0.7881

b) P(X < 42.8)

= P((X - mean)/sd < (42.8 - mean)/sd)

= P(Z < (42.8 - 50.89)/8.8)

= P(Z < -0.92)

= 0.1788

c) P(X > x) = 0.05

or, P((X - mean)/sd > (x - 50.89)/8.8) = 0.05

or, P(Z > (x - 50.89)/8.8) = 0.05

or, (x - 50.89)/8.8 = 0.95

or, x = 0.95 * 8.8 + 50.89

or, x = 59.25
d) P(X < x) = 0.75

or, P((X - mean)/sd < (x - 50.89)/8.8) = 0.75

or, P(Z < (x - 50.89)/8.8) = 0.75

or, (x - 50.89)/8.8) = 0.67

or, x = 0.67 * 8.8 + 50.89

or, x = 56.786

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