Assume that a set of test scores is normally distributed with a mean of 80 and a standard deviation of 25. Use the 68-95-99.7 rule to find the following quantities.

The percentage of scores less than 80 is?

To find the percentage of scores less than 80, we need to calculate the area under the normal distribution curve to the left of 80.

Since the data follows a normal distribution, we can use the 68-95-99.7 rule, also known as the empirical rule, which tells us that:
- 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, and
- Approximately 99.7% of the data falls within three standard deviations of the mean.

Since the mean is 80 and the standard deviation is 25, we need to calculate the area to the left of 80.

Using this information, we can say that:
- Approximately 68% of the data falls between 55 (80 - 25) and 105 (80 + 25).
- Therefore, approximately 34% of the data falls to the left of 80 (55 to 80).

So, the percentage of scores less than 80 is approximately 34%.

To find the percentage of scores less than 80, we need to calculate the area under the normal distribution curve to the left of the score 80. We can use the 68-95-99.7 rule to do this.

The 68-95-99.7 rule states that approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

Given that the mean is 80 and the standard deviation is 25, we can mark the three standard deviation boundaries on both sides of the mean:
- One standard deviation below the mean: 80 - 25 = 55
- Two standard deviations below the mean: 80 - 2(25) = 30
- Three standard deviations below the mean: 80 - 3(25) = -5

Since the scores cannot be less than zero in this case, we can ignore the third boundary and only consider the first two.

Now, we know that 68% falls within one standard deviation of the mean. So, the area to the left of score 80 is half of 68%, which is 34%.

Therefore, the percentage of scores less than 80 is approximately 34%.

Since 80 is the mean, the fraction le4ss than that is 50%. You don't need to invoke a 68-95-99.7 rule to figure that out. You also don't need to know the standard deviation.