Question

Park Rangers in a Yellowstone National Park have determined that fawns less than 6 months old have a body weight that is approximately normally distributed with a mean µ = 26.1 kg and standard deviation σ = 4.2 kg. Let x be the weight of a fawn in kilograms. Complete each of the following steps for the word problems below:  Rewrite each of the following word problems into a probability expression, such as P(x>30).  Convert each of the probability expressions involving x into probability expressions involving z, using the information from the scenario.  Sketch a normal curve for each z probability expression with the appropriate probability area shaded.  Solve the problem.

1. What is the probability of selecting a fawn less than 6 months old in Yellowstone that weighs less than 25 kilograms?

2. What is the probability of selecting a fawn less than 6 months old in Yellowstone that weighs more than 19 kilograms?

3. What is the probability of selecting a fawn less than 6 months old in Yellowstone that weighs between 30 and 38 kilograms?

4. If a fawn less than 6 months old weighs 16 pounds, would you say that it is an unusually small animal? Explain and verify your answer mathematically.

5. What is the weight of a fawn less than 6 months old that corresponds with a 20% probability of being randomly selected? Explain and verify your answer mathematically.

Answer 1

1) P(X < 25)

= P((X - )/ < (25 - )/)

= P(Z < (25 - 26.1)/4.2)

= P(Z < -0.26)

= 0.3974

2) P(X > 19)

= P((X - )/ > (19 - )/)

= P(Z > (19 - 26.1)/4.2)

= P(Z > -1.69)

= 1 - P(Z < -1.69)

= 1 - 0.0455

= 0.9454

3) P(30 < X < 38)

= P((30 - )/ < (X - )/ < (38 - )/)

= P((30 - 26.1)/4.2 < Z < (38 - 26.1)/4.2)

= P(0.93 < Z < 2.83)

= P(Z < 2.83) - P(Z < 0.93)

= 0.9977 - 0.8238

= 0.1739

4) P(X < 16)

= P((X - )/ < (16 - )/)

= P(Z < (16 - 26.1)/4.2)

= P(Z < -2.40)

= 0.0082

Since the probability is less than 0.05, so it is unusual.

5) P(X < x) = 0.20

or, P((X - )/ < (x - )/) = 0.20

or, P(Z < (x - 26.1)/4.2) = 0.20

or, (x - 26.1)/4.2 = -0.84

or, x = -0.84 * 4.2 + 26.1

or, x = 22.572