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
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between the two Z scores.
To find the approximate percentage of data values that is expected to fall between 54 and 66 in a bell-shaped distribution, we can use the empirical rule, also known as the 68-95-99.7 rule. This 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 this case, the mean is 60 and the standard deviation is 3.
To calculate the percentage of data values between 54 and 66, we need to determine how many standard deviations away from the mean these values are:
For 54:
(54 - 60) / 3 = -2
For 66:
(66 - 60) / 3 = 2
We can see that both 54 and 66 are 2 standard deviations away from the mean of 60.
Since the empirical rule tells us that approximately 95% of the data falls within two standard deviations of the mean, we can conclude that approximately 95% of the data values will fall between 54 and 66 in this bell-shaped distribution.
Therefore, the approximate percentage of data values expected to fall between 54 and 66 is 95%.