a) To find the percentage of quokkas that weigh over 14.4 pounds, we need to calculate the probability of the weight being greater than 14.4 in the Normal distribution N(10, 2.1).
Using a standard Normal distribution table or a calculator, we can convert the value of 14.4 to a Z-score using the formula: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
Z = (14.4 - 10) / 2.1 = 4.4 / 2.1 ≈ 2.0952
Now, we can look up the probability corresponding to a Z-score of 2.0952 in the standard Normal distribution table or use a calculator to find P(Z > 2.0952). This gives us the percentage of quokkas weighing over 14.4 pounds.
b) Similarly, to find the percentage of quokkas that weigh under 11.6 pounds, we calculate the probability of the weight being less than 11.6 in the Normal distribution N(10, 2.1). We can convert the value of 11.6 to a Z-score:
Z = (11.6 - 10) / 2.1 ≈ 1.0476
We then look up the probability corresponding to a Z-score of 1.0476, which gives us P(Z < 1.0476). This represents the percentage of quokkas weighing under 11.6 pounds.
c) To find the percentage of quokkas weighing between 11.6 and 14.4 pounds, we need to calculate the probability of the weight falling within that range. We can calculate the individual probabilities for each endpoint and subtract the lower probability from the higher probability.
P(11.6 < X < 14.4) = P(X < 14.4) - P(X < 11.6)
Using the Z-scores calculated earlier, we can look up the probabilities P(Z < 2.0952) and P(Z < 1.0476) in the standard Normal distribution table or use a calculator. Subtracting the lower probability from the higher probability gives us the percentage of quokkas weighing between 11.6 and 14.4 pounds.
Please note that the actual percentages will depend on the accuracy of the Z-scores and the precision used in the calculations.