Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.61 and a standard deviation of 0.43. Using the empirical rule, what percentage of the students have grade point averages that are at least 3.47? Please do not round your answer.
Well, to answer that question, all I need to do is juggle a few numbers for you!
According to the empirical rule (also known as the 68-95-99.7 rule), in a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean. That means 34% of the students have GPA scores between 2.61 - 0.43 and 2.61 + 0.43.
Alright, now let's move on to two standard deviations away from the mean. According to the rule, approximately 95% of the data falls within two standard deviations of the mean. So, we can say that 47% of the students have GPA scores between 2.61 - 2(0.43) and 2.61 + 2(0.43).
And finally, three standard deviations away from the mean. The empirical rule tells us that approximately 99.7% of the data falls within three standard deviations of the mean. So, we can infer that 49.7% of the students have GPA scores between 2.61 - 3(0.43) and 2.61 + 3(0.43).
Now, you specifically asked for the percentage of students with a GPA of at least 3.47, which is one standard deviation above the mean. Since we know that 34% of the students have a GPA within one standard deviation of the mean, we can subtract this number from 100% to find the percentage of students who have a GPA at least 3.47.
Therefore, the percentage of students with a GPA of at least 3.47 is approximately 100% - 34% = 66%.
To find the percentage of students with grade point averages that are at least 3.47, we need to determine the z-score for this value and use the empirical rule.
The z-score can be calculated using the formula:
z = (X - μ) / σ
where:
X is the value we want to convert to a z-score (in this case, 3.47),
μ is the mean (2.61), and
σ is the standard deviation (0.43).
Plugging in the values:
z = (3.47 - 2.61) / 0.43
z ≈ 2
Now, referring to the empirical rule, we know that approximately 95% of the data falls within 2 standard deviations of the mean (μ ± 2σ). Since our z-score is 2, it falls within this range.
Therefore, the percentage of students with grade point averages that are at least 3.47 is approximately 100% - 95% = 5%.
To find the percentage of students with grade point averages that are at least 3.47, we need to use the empirical rule, also known as the 68-95-99.7 rule.
According to the empirical rule, in a bell-shaped 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, we want to find the percentage of students with grade point averages that are at least 3.47. This value is greater than the mean, so we need to calculate the percentage outside of one standard deviation above the mean.
First, we need to find the z-score for a grade point average of 3.47. The z-score formula is given by:
z = (x - μ) / σ
Where:
- x is the value we want to find the z-score for (3.47 in this case)
- μ is the mean of the distribution (2.61 in this case)
- σ is the standard deviation of the distribution (0.43 in this case)
Using the formula:
z = (3.47 - 2.61) / 0.43 ≈ 2
This means that a grade point average of 3.47 is 2 standard deviations above the mean.
To find the percentage of students with grade point averages that are at least 3.47, we need to calculate the percentage of data beyond 2 standard deviations above the mean. Since the empirical rule tells us that approximately 95% of the data falls within two standard deviations of the mean, the remaining 5% is beyond two standard deviations.
Therefore, the percentage of students with grade point averages that are at least 3.47 is 5%.
Note: It is important to keep in mind that the empirical rule provides approximate percentages and assumes a perfectly normal distribution. In practice, actual distributions may deviate slightly from the normal distribution.