Observations are drawn from a bell-shaped distribution with a mean of 35 and a standard deviation of 2.
a. Approximately what percentage of the observations fall between 33 and 37? (Round your answer to the nearest whole percent.)
b. Approximately what percentage of the observations fall between 31 and 39? (Round your answer to the nearest whole percent.)
c. Approximately what percentage of the observations are less than 33? (Round your answer to 1 decimal place.)
a. To find the percentage of observations that fall between 33 and 37, we need to find the area under the bell curve corresponding to this range. We can use the empirical rule which states that approximately 68% of the data falls within one standard deviation of the mean. Since the standard deviation is 2, this means that approximately 34% of the data falls between 35-2=33 and 35+2=37. So, approximately 34% of the observations fall between 33 and 37.
b. Similarly, since 31 is two standard deviations below the mean and 39 is two standard deviations above the mean, approximately 95% of the data falls within this range according to the empirical rule. So, approximately 95% of the observations fall between 31 and 39.
c. To find the percentage of observations that are less than 33, we need to find the area under the bell curve to the left of 33. Using a standard normal distribution table or a calculator, we find that the area to the left of 33 is approximately 0.1587 or 15.9% when rounded to 1 decimal place. So, approximately 15.9% of the observations are less than 33.
Observations are drawn from a bell-shaped distribution with a mean of 210 and a standard deviation of 5. There are 1,900 observations in the data set.
a. Approximately what percentage of the observations are less than 220? (Round your answer to 1 decimal place.)
b. Approximately how many observations are less than 220? (Round your answer to the nearest whole number.)
To solve these questions, we can use the empirical rule, also known as the 68-95-99.7 rule, which states that for a bell-shaped 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 that fall between 33 and 37, we will calculate the z-scores for these values and use the empirical rule:
z-score for 33 = (33 - 35) / 2 = -1
z-score for 37 = (37 - 35) / 2 = 1
Using the empirical rule, approximately 68% of the observations fall within 1 standard deviation of the mean. Therefore, approximately 34% (half of 68%) of the observations fall between each z-score.
So, the percentage of observations that fall between 33 and 37 is approximately 34%.
b. To find the percentage of observations that fall between 31 and 39, we will use the same approach:
z-score for 31 = (31 - 35) / 2 = -2
z-score for 39 = (39 - 35) / 2 = 2
Using the empirical rule, approximately 95% of the observations fall within 2 standard deviations of the mean. Therefore, approximately 47.5% (half of 95%) of the observations fall between each z-score.
So, the percentage of observations that fall between 31 and 39 is approximately 95%.
c. To find the percentage of observations that are less than 33, we will calculate the z-score for 33 and use a z-table to find the corresponding percentage:
z-score for 33 = (33 - 35) / 2 = -1
Using a z-table, we can find that approximately 34.1% of the observations are less than a z-score of -1.
So, the percentage of observations that are less than 33 is approximately 34.1%.
a. To find the percentage of observations that are less than 220, we need to find the area under the bell curve to the left of 220. We can standardize the value 220 using the z-score formula:
z = (x - μ) / σ
Where x is the value we want to standardize, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:
z = (220 - 210) / 5
z = 2
Using a standard normal distribution table or a calculator, we find that the area to the left of 2 is approximately 0.9772 or 97.7% when rounded to 1 decimal place. So, approximately 97.7% of the observations are less than 220.
b. To find the number of observations that are less than 220, we can multiply the percentage we found in part a by the total number of observations in the data set:
Number of observations = Percentage * Total number of observations
Number of observations = 0.9772 * 1900
Number of observations ≈ 1836
Round to the nearest whole number gives us approximately 1836 observations that are less than 220.