Observations are drawn from a bell-shaped distribution with a mean of 35 and a standard deviation of 3.
a. Approximately what percentage of the observations fall between 32 and 38? (Round your answer to the nearest whole percent.)
b. Approximately what percentage of the observations fall between 29 and 41? (Round your answer to the nearest whole percent.)
a. To find the percentage of observations that fall between 32 and 38, we need to find the area under the bell curve between these two values.
First, we need to standardize the values by using the z-score formula:
z = (x - mean) / standard deviation
For 32:
z1 = (32 - 35) / 3 = -1
For 38:
z2 = (38 - 35) / 3 = 1
Using a standard normal distribution table or a calculator, we can find the percentage of observations between -1 and 1.
The area to the left of z1 is 0.1587 (from the standard normal distribution table) and the area to the left of z2 is 0.8413.
Therefore, the percentage of observations between 32 and 38 is approximately:
0.8413 - 0.1587 = 0.6826
To convert this to a percentage, we multiply by 100:
0.6826 * 100 ≈ 68%
b. Using the same process, we can find the percentage of observations that fall between 29 and 41.
For 29:
z1 = (29 - 35) / 3 = -2
For 41:
z2 = (41 - 35) / 3 = 2
Using the standard normal distribution table, the area to the left of z1 is 0.0228 and the area to the left of z2 is 0.9772.
Therefore, the percentage of observations between 29 and 41 is approximately:
0.9772 - 0.0228 = 0.9544
Converting this to a percentage:
0.9544 * 100 ≈ 95%
So, approximately 95% of the observations fall between 29 and 41.
To find the percentage of observations that fall between two values in a bell-shaped distribution, we can use the Empirical Rule. The Empirical Rule states that in a normal distribution:
- Approximately 68% of the observations fall within one standard deviation of the mean
- Approximately 95% of the observations fall within two standard deviations of the mean
- Approximately 99.7% of the observations fall within three standard deviations of the mean
a. To find the percentage of observations between 32 and 38, we need to determine the number of standard deviations away from the mean each value is.
For 32: (32 - 35) / 3 = -1
For 38: (38 - 35) / 3 = 1
Since the values are 1 standard deviation away from the mean, we can use the Empirical Rule and conclude that approximately 68% of the observations fall between 32 and 38.
b. To find the percentage of observations between 29 and 41, we repeat the same steps:
For 29: (29 - 35) / 3 = -2
For 41: (41 - 35) / 3 = 2
Since the values are 2 standard deviations away from the mean, we can use the Empirical Rule and conclude that approximately 95% of the observations fall between 29 and 41.
Therefore, the answers are:
a. Approximately 68% of the observations fall between 32 and 38.
b. Approximately 95% of the observations fall between 29 and 41.