Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.62 and a standard deviation of 0.43. Using the empirical rule, what percentage of the students have grade point averages that are at least 3.48? Please do not round your answer.
To answer this question, we need to use the empirical rule for a normal distribution. According to the empirical rule, approximately:
- 68% of the values fall within one standard deviation of the mean
- 95% of the values fall within two standard deviations of the mean
- 99.7% of the values fall within three standard deviations of the mean
Given that the mean is 2.62 and the standard deviation is 0.43, we can calculate the value of 3.48 in terms of standard deviations from the mean.
First, we calculate the difference between the value and the mean: 3.48 - 2.62 = 0.86.
Next, we divide the difference by the standard deviation to get the number of standard deviations: 0.86 / 0.43 = 2.
Since 3.48 is 2 standard deviations above the mean, we can use the empirical rule to determine the percentage of values that fall at or above this value. According to the empirical rule, approximately 2.5% of the values fall beyond two standard deviations above the mean.
Therefore, the percentage of students with grade point averages that are at least 3.48 is approximately 2.5%.
To calculate the percentage of students with grade point averages at least 3.48 using the empirical rule, we need to find the z-score associated with that value.
The z-score measures the number of standard deviations an observation is away from the mean. We can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- x is the value we want to find the z-score for (in this case, 3.48)
- μ is the mean of the distribution (2.62)
- σ is the standard deviation of the distribution (0.43)
Plugging in the values we have, we get:
z = (3.48 - 2.62) / 0.43
z = 1.994
Now that we have the z-score, we can use the empirical rule to determine the percentage of students with grade point averages at least 3.48.
According to the empirical rule, for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
In this case, we are interested in the percentage of students with grade point averages greater than 3.48, which is beyond 1 standard deviation from the mean.
Since the z-score is 1.994, which is almost 2 (greater than 1 but less than 2), we can estimate that approximately 95% of the students have grade point averages at least 3.48 using the empirical rule.
Therefore, the percentage of students with grade point averages at least 3.48 is approximately 95%.
3.48 - 2.62 = .86 ... 2 sd above the mean
look at the percentages on the bell curve