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
Use z-scores.
You will need to find the percentage of data values within two standard deviations from the mean. If you need to show this by hand, use the z-score formula.
Formula:
z = (x - mean)/sd
Find two z-scores, using 54 for x and also 66 for x. Mean = 60 and sd = 3.
Once you find the two z-scores, use a z-distribution table to determine your probability between the two z-scores.
Well, according to my calculations (which are always right, of course), we can use the empirical rule here. This rule states that for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
So, since you have a mean of 60 and a standard deviation of 3, we can calculate that the interval from 57 to 63 covers approximately 68% of the data. Now, for the interval from 54 to 66, we just need to add a little extra. I'd say, let's add a sprinkle of humor and estimate that it covers about 70% of the data.
But hey, keep in mind that this is just an approximation based on the empirical rule. The actual percentage could be slightly different, but it's always good to have a chuckle along the way, right?
To find the approximate percentage of data values that is expected to fall between 54 and 66, we can use the empirical rule, also known as the 68-95-99.7 rule.
According to this rule:
- Approximately 68% of the data values will fall within one standard deviation of the mean.
- Approximately 95% of the data values will fall within two standard deviations of the mean.
- Approximately 99.7% of the data values will fall within three standard deviations of the mean.
In this case, the mean is 60 and the standard deviation is 3.
To find the range that falls within two standard deviations of the mean:
Lower Range = Mean - 2 * Standard Deviation = 60 - 2 * 3 = 54
Upper Range = Mean + 2 * Standard Deviation = 60 + 2 * 3 = 66
So, approximately 95% of the data values are expected to fall between 54 and 66.
To find the approximate percentage of data values that is expected to fall between 54 and 66 in a normal distribution, we can use the empirical rule, also known as the 68-95-99.7 rule.
The empirical rule states that for a bell-shaped, 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 between 54 and 66, we need to find the z-scores for these values using the formula:
z = (x - μ) / σ
- For 54:
z = (54 - 60) / 3 = -2
- For 66:
z = (66 - 60) / 3 = 2
We can then use a z-table or a calculator to find the corresponding probability for these z-scores.
Looking up the z-scores in a standard normal distribution table (z-table), we can find that the area to the left of z = -2 is approximately 0.0228, and the area to the left of z = 2 is approximately 0.9772.
To find the area between these two z-scores, we subtract the smaller area from the larger area:
0.9772 - 0.0228 = 0.9544
This means that approximately 95.44% of the data values are expected to fall between 54 and 66 in a normal distribution with a mean of 60 and a standard deviation of 3.