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### c. What is the Z-score for a giraffe that is 18.5 feet tall?
The Z-score formula is used to determine how many standard deviations away a specific value is from the mean. It is given by:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\( X \)[/tex] is the value of interest (in this case, the giraffe height of 18.5 feet),
- [tex]\( \mu \)[/tex] is the mean height (which is 18 feet),
- [tex]\( \sigma \)[/tex] is the standard deviation (which is 1.5 feet).
Using these values:
[tex]\[ Z = \frac{18.5 - 18}{1.5} = \frac{0.5}{1.5} \approx 0.333 \][/tex]
So, the Z-score for a giraffe that is 18.5 feet tall is approximately 0.333.
### d. What is the probability that a randomly selected giraffe will be shorter than 17.3 feet tall?
To find this probability, we first need to calculate the Z-score for a giraffe that is 17.3 feet tall:
[tex]\[ Z = \frac{17.3 - 18}{1.5} = \frac{-0.7}{1.5} \approx -0.467 \][/tex]
Next, we look up this Z-score in the standard normal distribution table or use a cumulative distribution function (CDF). The cumulative probability for a Z-score of approximately -0.467 is about 0.3204.
So, the probability that a randomly selected giraffe will be shorter than 17.3 feet tall is approximately 0.3204 (or 32.04%).
### e. What is the probability that a randomly selected giraffe will be between 17.2 and 18.2 feet tall?
To find this probability, we need to calculate the Z-scores for 17.2 feet and 18.2 feet.
For 17.2 feet:
[tex]\[ Z_{\text{lower}} = \frac{17.2 - 18}{1.5} = \frac{-0.8}{1.5} \approx -0.533 \][/tex]
For 18.2 feet:
[tex]\[ Z_{\text{upper}} = \frac{18.2 - 18}{1.5} = \frac{0.2}{1.5} \approx 0.133 \][/tex]
Next, we look up these Z-scores in the standard normal distribution table or use a CDF. The cumulative probabilities for these Z-scores are:
- For [tex]\( Z_{\text{lower}} \approx -0.533 \)[/tex], the cumulative probability is approximately 0.297.
- For [tex]\( Z_{\text{upper}} \approx 0.133 \)[/tex], the cumulative probability is approximately 0.553.
The probability that a randomly selected giraffe will be between 17.2 and 18.2 feet tall is the difference between these cumulative probabilities:
[tex]\[ P(17.2 < X < 18.2) = P(Z_{\text{upper}}) - P(Z_{\text{lower}}) = 0.553 - 0.297 \approx 0.256 \][/tex]
So, the probability that a randomly selected giraffe will be between 17.2 and 18.2 feet tall is approximately 0.256 (or 25.6%).
By following these steps and using the normal distribution properties, we can adequately answer each sub-question.
### c. What is the Z-score for a giraffe that is 18.5 feet tall?
The Z-score formula is used to determine how many standard deviations away a specific value is from the mean. It is given by:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\( X \)[/tex] is the value of interest (in this case, the giraffe height of 18.5 feet),
- [tex]\( \mu \)[/tex] is the mean height (which is 18 feet),
- [tex]\( \sigma \)[/tex] is the standard deviation (which is 1.5 feet).
Using these values:
[tex]\[ Z = \frac{18.5 - 18}{1.5} = \frac{0.5}{1.5} \approx 0.333 \][/tex]
So, the Z-score for a giraffe that is 18.5 feet tall is approximately 0.333.
### d. What is the probability that a randomly selected giraffe will be shorter than 17.3 feet tall?
To find this probability, we first need to calculate the Z-score for a giraffe that is 17.3 feet tall:
[tex]\[ Z = \frac{17.3 - 18}{1.5} = \frac{-0.7}{1.5} \approx -0.467 \][/tex]
Next, we look up this Z-score in the standard normal distribution table or use a cumulative distribution function (CDF). The cumulative probability for a Z-score of approximately -0.467 is about 0.3204.
So, the probability that a randomly selected giraffe will be shorter than 17.3 feet tall is approximately 0.3204 (or 32.04%).
### e. What is the probability that a randomly selected giraffe will be between 17.2 and 18.2 feet tall?
To find this probability, we need to calculate the Z-scores for 17.2 feet and 18.2 feet.
For 17.2 feet:
[tex]\[ Z_{\text{lower}} = \frac{17.2 - 18}{1.5} = \frac{-0.8}{1.5} \approx -0.533 \][/tex]
For 18.2 feet:
[tex]\[ Z_{\text{upper}} = \frac{18.2 - 18}{1.5} = \frac{0.2}{1.5} \approx 0.133 \][/tex]
Next, we look up these Z-scores in the standard normal distribution table or use a CDF. The cumulative probabilities for these Z-scores are:
- For [tex]\( Z_{\text{lower}} \approx -0.533 \)[/tex], the cumulative probability is approximately 0.297.
- For [tex]\( Z_{\text{upper}} \approx 0.133 \)[/tex], the cumulative probability is approximately 0.553.
The probability that a randomly selected giraffe will be between 17.2 and 18.2 feet tall is the difference between these cumulative probabilities:
[tex]\[ P(17.2 < X < 18.2) = P(Z_{\text{upper}}) - P(Z_{\text{lower}}) = 0.553 - 0.297 \approx 0.256 \][/tex]
So, the probability that a randomly selected giraffe will be between 17.2 and 18.2 feet tall is approximately 0.256 (or 25.6%).
By following these steps and using the normal distribution properties, we can adequately answer each sub-question.
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