Weights of male mountain lions follow the normal distribution with a median of 150 lb and an interquartile range of 8.2 lb.
Find the 75th percentile of the weights.
Find the 95th percentile of the weights.
FOR THE 1ST THE ANSWER IS 154.1
the 2nd the answer is 159.95
why 159.95?
To find the percentiles of the weights, we need to use the properties of the normal distribution. We know that the normal distribution can be fully described by its mean and standard deviation. However, in this case, we are given the median and interquartile range (IQR), which is a measure of the spread of the distribution.
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1), so we have:
IQR = Q3 - Q1
Given the IQR, we can find the standard deviation using the formula:
IQR = 1.349 * σ
where σ is the standard deviation. Dividing both sides of the equation by 1.349, we get:
σ = IQR / 1.349
Now, we can find the percentiles using the median (Q2), mean (μ), and standard deviation (σ). In a normal distribution, the median (Q2) and mean (μ) are the same.
To find the 75th percentile:
- Calculate the z-score for the desired percentile using the standard normal distribution table or a calculator. For the 75th percentile, the z-score is approximately 0.674.
- Use the formula:
X = μ + (z * σ)
where X is the weight at the desired percentile, μ is the median/mean, z is the z-score, and σ is the standard deviation.
To find the 95th percentile:
- Calculate the z-score for the desired percentile using the standard normal distribution table or a calculator. For the 95th percentile, the z-score is approximately 1.645.
- Use the formula:
X = μ + (z * σ)
where X is the weight at the desired percentile, μ is the median/mean, z is the z-score, and σ is the standard deviation.
Applying these steps to the given information:
- The median (μ) is 150 lb.
- The interquartile range (IQR) is 8.2 lb.
- Calculating the standard deviation (σ):
σ = IQR / 1.349
σ = 8.2 lb / 1.349
σ ≈ 6.07 lb
Now, we can find the percentiles:
To find the 75th percentile:
- Using the formula X = μ + (z * σ):
X = 150 lb + (0.674 * 6.07 lb)
X ≈ 154.09 lb
Therefore, the 75th percentile of the weights is approximately 154.09 lb.
To find the 95th percentile:
- Using the formula X = μ + (z * σ):
X = 150 lb + (1.645 * 6.07 lb)
X ≈ 159.93 lb
Therefore, the 95th percentile of the weights is approximately 159.93 lb.