Data with 150 observations are drawn from a bell-shaped distribution with a mean of 30 and a standard deviation of 7. Approximately how many observations are more than 44? (Round your answer to the nearest whole number.)
To solve this problem, 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
Since the mean is 30 and the standard deviation is 7, we can calculate the number of observations that fall within the range of 44 or higher by calculating the z-score and using the empirical rule.
First, we calculate the z-score for 44:
z = (x - μ) / σ
z = (44 - 30) / 7
z = 2
According to the empirical rule, approximately 95% of the observations fall within two standard deviations of the mean. This means that approximately 5% of the observations fall beyond two standard deviations from the mean.
To calculate the number of observations that are more than 44, we need to calculate the remaining 5% of the observations. We can calculate this using the area under the standard normal curve, which is available in the standard normal distribution table.
Using the standard normal distribution table, we find that the area to the left of z = 2 is approximately 0.9772. This means that the remaining 1 - 0.9772 = 0.0228, or approximately 2.28%, of the observations fall beyond z = 2.
To find the number of observations that corresponds to this percentage, we multiply the number of observations (150) by the percentage (0.0228):
Number of observations = 150 * 0.0228 = 3.42
Rounding to the nearest whole number, we find that approximately 3 observations are more than 44.
To answer this question, we can use the properties of the normal distribution. Given that the data follows a bell-shaped distribution with a mean of 30 and a standard deviation of 7, we can calculate the z-score for the value of 44 using the formula:
z = (x - μ) / σ
where:
z is the z-score,
x is the value we want to find the probability for,
μ is the mean of the distribution,
σ is the standard deviation of the distribution.
Plugging in the values for this problem:
x = 44,
μ = 30,
σ = 7.
z = (44 - 30) / 7
z = 14 / 7
z = 2
Now, we can use a standard normal distribution table to find the probability associated with the z-score of 2. A z-score of 2 corresponds to a probability of approximately 0.9772.
Since we are interested in the number of observations that are more than 44, we need to find the probability of getting a value greater than 44. This is equivalent to subtracting the cumulative probability from 1.
1 - 0.9772 = 0.0228
So, approximately 0.0228 or 2.28% of the observations are more than 44.
Finally, to find the number of observations, we can multiply the probability by the total number of observations:
0.0228 * 150 = 3.42
Rounding this to the nearest whole number, we can conclude that approximately 3 observations are more than 44.