Given that a sample is approximately bell-shaped with a mean of 60 and a standard deviation of 3, the approximate percentage of data values that is expected to fall between 54 and 66 is
75%
95%
68%
99.7%
99.7
To find the approximate percentage of data values that is expected to fall between 54 and 66 in a bell-shaped distribution, you can use the empirical rule.
The empirical rule states that for a normal distribution:
- Approximately 68% of the data values fall within one standard deviation of the mean.
- Approximately 95% of the data values fall within two standard deviations of the mean.
- Approximately 99.7% of the data values fall within three standard deviations of the mean.
In this case, the mean is 60 and the standard deviation is 3.
To calculate the percentage of data values that fall between 54 and 66, we can calculate the Z-scores for these values and find the area under the standard normal curve between those Z-scores.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Observed value
μ = Mean
σ = Standard deviation
For the lower bound (54), the Z-score is:
Z1 = (54 - 60) / 3 = -2
For the upper bound (66), the Z-score is:
Z2 = (66 - 60) / 3 = 2
Now, we can use a standard normal distribution table or a statistical calculator to find the area under the curve between Z1 and Z2.
Using the standard normal distribution table, we find that the area between Z = -2 and Z = 2 is approximately 0.9545.
Therefore, the approximate percentage of data values that is expected to fall between 54 and 66 is 95%.
So, the answer is:
95%