Answer :
To find the percentage of goldfish that have lengths between 4.6 inches and 9.4 inches, we will use the mean and the standard deviation provided in the table.
1. Identify the given values:
- Mean length of goldfish (\(\mu\)) = 7 inches
- Standard deviation (\(\sigma\)) = 1.2 inches
- Lower bound length = 4.6 inches
- Upper bound length = 9.4 inches
2. Calculate the z-scores for the lower and upper bounds:
The z-score for any length \(X\) is given by the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
- For the lower bound (4.6 inches):
[tex]\[ Z_{\text{lower}} = \frac{4.6 - 7}{1.2} = \frac{-2.4}{1.2} = -2 \][/tex]
- For the upper bound (9.4 inches):
[tex]\[ Z_{\text{upper}} = \frac{9.4 - 7}{1.2} = \frac{2.4}{1.2} = 2 \][/tex]
3. Interpret the z-scores using the empirical rule (68-95-99.7 rule):
- About 68% of data falls within 1 standard deviation from the mean (\(-1 \leq Z \leq 1\)).
- About 95% of data falls within 2 standard deviations from the mean (\(-2 \leq Z \leq 2\)).
- About 99.7% of data falls within 3 standard deviations from the mean (\(-3 \leq Z \leq 3\)).
Since both calculated z-scores \(Z_{\text{lower}} = -2\) and \(Z_{\text{upper}} = 2\) are within 2 standard deviations from the mean, this falls into the 95% range.
However, they also fall exactly at the boundaries of the 99.7% range (since \(-2\) and \(2\) are within \(-3\) to \(3\) standard deviations range as well).
Hence, the percentage of goldfish with lengths between 4.6 inches and 9.4 inches is [tex]\(99.7 \%.\)[/tex]
1. Identify the given values:
- Mean length of goldfish (\(\mu\)) = 7 inches
- Standard deviation (\(\sigma\)) = 1.2 inches
- Lower bound length = 4.6 inches
- Upper bound length = 9.4 inches
2. Calculate the z-scores for the lower and upper bounds:
The z-score for any length \(X\) is given by the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
- For the lower bound (4.6 inches):
[tex]\[ Z_{\text{lower}} = \frac{4.6 - 7}{1.2} = \frac{-2.4}{1.2} = -2 \][/tex]
- For the upper bound (9.4 inches):
[tex]\[ Z_{\text{upper}} = \frac{9.4 - 7}{1.2} = \frac{2.4}{1.2} = 2 \][/tex]
3. Interpret the z-scores using the empirical rule (68-95-99.7 rule):
- About 68% of data falls within 1 standard deviation from the mean (\(-1 \leq Z \leq 1\)).
- About 95% of data falls within 2 standard deviations from the mean (\(-2 \leq Z \leq 2\)).
- About 99.7% of data falls within 3 standard deviations from the mean (\(-3 \leq Z \leq 3\)).
Since both calculated z-scores \(Z_{\text{lower}} = -2\) and \(Z_{\text{upper}} = 2\) are within 2 standard deviations from the mean, this falls into the 95% range.
However, they also fall exactly at the boundaries of the 99.7% range (since \(-2\) and \(2\) are within \(-3\) to \(3\) standard deviations range as well).
Hence, the percentage of goldfish with lengths between 4.6 inches and 9.4 inches is [tex]\(99.7 \%.\)[/tex]
