suppose that grade point averages of undergraduate students at one university have a bell shaped distribution with a mean of 2.55 and a standard deviation of 0.43. Using the empirical rule, what percentage of students have grade point averages that are between 1.26 and 3.84

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.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores.

To find the percentage of students with grade point averages between 1.26 and 3.84, we can use the empirical rule, which states that for 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.

Step 1: Calculate the z-scores for the given grade point averages.
To calculate the z-scores, we use the formula: z = (x - μ) / σ
where z is the z-score, x is the value, μ is the mean, and σ is the standard deviation.

For 1.26, z = (1.26 - 2.55) / 0.43 = -3
For 3.84, z = (3.84 - 2.55) / 0.43 = 3

Step 2: Determine the percentage of data within three standard deviations of the mean.
Since the value of -3 corresponds to approximately the lower end of the data and the value of 3 corresponds to approximately the upper end of the data, we can use the empirical rule to calculate the percentage of data within these z-score values.

Approximately 99.7% of the data falls within three standard deviations of the mean. Therefore, the percentage of data between -3 and 3 is 99.7%.

Step 3: Calculate the percentage of data between 1.26 and 3.84.
Since we know that 99.7% of the data falls within three standard deviations of the mean, and we want to find the percentage between 1.26 and 3.84, we need to subtract the percentage of data outside this range (beyond three standard deviations) from 99.7%.

Since the empirical rule states that 99.7% of the data falls within three standard deviations of the mean, we can subtract this from 100% to find the percentage outside the range. Therefore, the percentage of data outside the range -3 to 3 is 100% - 99.7% = 0.3%.

However, we want to find the percentage between 1.26 and 3.84, so we need to subtract the percentage outside the range of 1.26 to the percentage outside the range of 3.84.

Since the bell curve is symmetric, the percentages outside these ranges would be the same. Therefore, we can divide 0.3% by 2 to get the percentage of data outside the range of either 1.26 or 3.84.

0.3% / 2 = 0.15%.

Finally, to find the percentage of data between 1.26 and 3.84, we subtract this calculated percentage of data outside the range from 100%.

100% - 0.15% = 99.85%.

Therefore, approximately 99.85% of students have grade point averages between 1.26 and 3.84.

To find the percentage of students with grade point averages between 1.26 and 3.84 using the empirical rule, we need to calculate the z-scores for these values. The z-score measures how many standard deviations a data point is away from the mean.

The formula to calculate the z-score is:
z = (x - μ) / σ

where:
z = z-score
x = data point
μ = mean
σ = standard deviation

For the lower value, 1.26:
z1 = (1.26 - 2.55) / 0.43

For the upper value, 3.84:
z2 = (3.84 - 2.55) / 0.43

Now we can find the corresponding percentage using the z-table or a statistical calculator that provides the cumulative probability. The cumulative probability gives the area under the normal distribution curve up to a certain z-score.

Using a z-table or calculator, we can find the cumulative probability for z1 and z2. Let's assume the cumulative probabilities are P1 and P2, respectively.

The percent of students with grade point averages between 1.26 and 3.84 is the difference between P2 and P1, multiplied by 100%:
Percentage = (P2 - P1) * 100%

Please note that the empirical rule assumes a normal distribution, and this approximation may not be perfectly accurate in practice.