How do I calculate the sample data that falls within the interval? It says from the interval 425.5 - 454.5
This should be 95% of the population data within the interval (according to empirical rule). The question states I SHOULD expect 95%, but the answer is 92.22%. Why is it 92.22 and not 95? how do I calculate this?
*also I understand I did not provide the table of values as it was really large, I just want a rough estimate on how could I make this calculation. I've been stuck on it for a few hours.
To calculate the percentage of sample data that falls within a given interval, you need to use the empirical rule or the normal distribution.
The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In your case, the interval given is 425.5 - 454.5. To calculate the percentage of sample data that falls within this interval, you need to follow these steps:
1. Determine the mean and standard deviation of your sample data. If you have access to the entire dataset, you can calculate these directly. If you only have summary statistics or a large dataset, you can use statistical software or an online calculator to estimate them.
2. Calculate the z-scores for the lower and upper bounds of your interval. The z-score is given by the formula: z = (x - μ) / σ, where x is the specific value, μ is the mean, and σ is the standard deviation.
For the lower bound (425.5):
z_lower = (425.5 - μ) / σ
For the upper bound (454.5):
z_upper = (454.5 - μ) / σ
3. Look up the corresponding values in the standard normal distribution table or use a z-score calculator. These tables or calculators give you the proportion of the data that falls below a given z-score.
4. Calculate the percentage of the data between the two z-scores. Subtract the lower z-score from the upper z-score and multiply by 100.
Percentage = (upper_z - lower_z) * 100
While the empirical rule suggests 95% of the data should fall within the interval, the actual percentage may differ slightly due to sample size or other factors. Calculating the exact percentage requires the use of the normal distribution table or calculator.
Note: Keep in mind that these calculations assume a normal distribution. If your data does not meet this assumption, alternative methods may need to be used.