To determine what percentage of goldfish have lengths between 4.6 inches and 9.4 inches, we need to analyze this range in the context of the given mean and standard deviation. Here's the step-by-step solution:
1. Determine the Mean and Standard Deviation:
- The mean length of the goldfish is 7 inches.
- The standard deviation of the goldfish length is 1.2 inches.
2. Calculate the Z-Scores for the given bounds:
- The lower bound of the length is 4.6 inches.
- The upper bound of the length is 9.4 inches.
The formula to calculate the z-score is:
[tex]\[
z = \frac{(X - \text{mean})}{\text{standard deviation}}
\][/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):
- The empirical rule states that for a normal distribution:
- Approximately 68% of data falls within 1 standard deviation of the mean.
- Approximately 95% of data falls within 2 standard deviations of the mean.
- Approximately 99.7% of data falls within 3 standard deviations of the mean.
In this case, we have z-scores of -2 and 2, which correspond to 2 standard deviations from the mean. According to the empirical rule, 95% of the data falls within this range.
Therefore, the percentage of goldfish with lengths between 4.6 inches and 9.4 inches is approximately [tex]\( \boxed{95\%} \)[/tex].
