Answer :
Given the information provided, let's solve each of the questions step-by-step.
1. What is the distribution of \(X\)?
Since the height of an adult giraffe \(X\) is normally distributed with a mean of 17 feet and a standard deviation of 0.8 feet, we can denote this distribution as:
[tex]\[ X \sim N(17, 0.8) \][/tex]
2. What is the median giraffe height?
In a normal distribution, the mean is equal to the median. Therefore, the median height of the giraffes is:
[tex]\[ 17 \, \text{ft} \][/tex]
3. What is the \(z\)-score for a giraffe that is 19 feet tall?
The \(z\)-score is calculated as follows:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Plugging in the values:
[tex]\[ z = \frac{19 - 17}{0.8} = \frac{2}{0.8} = 2.5 \][/tex]
4. What is the probability that a randomly selected giraffe will be shorter than 17.4 feet tall?
To find this probability, we use the cumulative distribution function (CDF) for a normal distribution. The probability is:
[tex]\[ P(X < 17.4) = 0.6915 \][/tex]
5. What is the probability that a randomly selected giraffe will be between 18 and 18.8 feet tall?
To find this probability, we calculate the CDF for both bounds and subtract the results:
[tex]\[ P(18 < X < 18.8) = P(X < 18.8) - P(X < 18) = 0.0934 \][/tex]
6. The 80th percentile for the height of giraffes is:
The 80th percentile of a normal distribution can be found using the inverse of the cumulative distribution function (also known as the percent-point function or quantile function):
[tex]\[ 80\text{th percentile} = 17.6733 \, \text{ft} \][/tex]
By following these steps, we have addressed each part of the problem accurately.
1. What is the distribution of \(X\)?
Since the height of an adult giraffe \(X\) is normally distributed with a mean of 17 feet and a standard deviation of 0.8 feet, we can denote this distribution as:
[tex]\[ X \sim N(17, 0.8) \][/tex]
2. What is the median giraffe height?
In a normal distribution, the mean is equal to the median. Therefore, the median height of the giraffes is:
[tex]\[ 17 \, \text{ft} \][/tex]
3. What is the \(z\)-score for a giraffe that is 19 feet tall?
The \(z\)-score is calculated as follows:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Plugging in the values:
[tex]\[ z = \frac{19 - 17}{0.8} = \frac{2}{0.8} = 2.5 \][/tex]
4. What is the probability that a randomly selected giraffe will be shorter than 17.4 feet tall?
To find this probability, we use the cumulative distribution function (CDF) for a normal distribution. The probability is:
[tex]\[ P(X < 17.4) = 0.6915 \][/tex]
5. What is the probability that a randomly selected giraffe will be between 18 and 18.8 feet tall?
To find this probability, we calculate the CDF for both bounds and subtract the results:
[tex]\[ P(18 < X < 18.8) = P(X < 18.8) - P(X < 18) = 0.0934 \][/tex]
6. The 80th percentile for the height of giraffes is:
The 80th percentile of a normal distribution can be found using the inverse of the cumulative distribution function (also known as the percent-point function or quantile function):
[tex]\[ 80\text{th percentile} = 17.6733 \, \text{ft} \][/tex]
By following these steps, we have addressed each part of the problem accurately.
