Answer :
The given confidence interval is μ ± 20, which is a range of weights between 70 - 20 = 50 kg and 120 + 20 = 140 kg. Since the weight of the giant panda is assumed to be Normally distributed.
The mean μ can be found by taking the midpoint of the given range:μ = (70 + 120)/2 = 95 kgThe standard deviation σ can be found by using the fact that the given range is the μ ± 20 confidence interval. In other words,20 = zσwhere z is the z-score corresponding to the desired level of confidence.
For a 95% confidence interval, z = 1.96 (from standard normal table).Therefore,σ = 20/1.96 = 10.2 kg.Now we want to find the proportion of giant pandas that weigh between 100.5 and 103.25 kg, which can be expressed in terms of z-scores:z1 = (100.5 - μ)/σ = (100.5 - 95)/10.2 ≈ 0.49z2 = (103.25 - μ)/σ = (103.25 - 95)/10.2 ≈ 0.81Using a standard normal table or calculator, we can find the proportion of the area under the curve between these z-scores:P(0.49 < Z < 0.81) ≈ 0.1386Therefore, the proportion of giant pandas that weigh between 100.5 and 103.25 kg is approximately 0.1386, rounded to four decimal places. So, the answer is 0.1386.
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Final answer:
The question requires the conversion of weights to Z-scores, and the use of a standard Normal distribution table or statistical software to compute the proportion of giant pandas between these weights. However, without these tools or values at hand, we cannot find the precise proportion.
Explanation:
The question asks what proportion of giant pandas weigh between 100.5 and 103.25 kg if their weight distribution is Normal and the given range is the μ ± 2σ confidence interval. From the given range, we can infer that the mean, μ is 70-120/2 = 95 kg and the standard deviation, σ, is (120-70)/4 = 12.5 kg.
Next, we need to convert the weights 100.5kg and 103.25kg to Z-scores, which are measures of how many standard deviations an element is from the mean. The formula to calculate Z-score is: Z = (X - μ)/σ.
Calculation of Z-scores
For X = 100.5kg, Z1 = (100.5kg - 95kg) / 12.5kg = 0.44.
For X = 103.25kg, Z2 = (103.25kg - 95kg) / 12.5kg = 0.66.
The proportion of giant pandas between these weights is represented by the area under the Normal curve between these Z-scores. This can be found by looking up the Z-scores in a standard Normal distribution table, or by using a statistics calculator or software. In this case, area between Z1 and Z2 will give the proportion which generally needs to be calculated by statistical software. Unfortunately, without these tools or values at the moment, we cannot proceed to compute the exact decimal number.
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