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.
