If your score on a statistics exam was 76 and the professor gave you the distribution for the exam score for your class, you could find your percentile to understand where you stand in comparison to your fellow students. Assume that the distribution for exam scores is normally distributed with a mean of 80 and a standard deviation of 8.

a) What is the percentile corresponding to your test score?
b) In terms of percentiles, would you be better off in a class for which scores were normally distributed with a mean of 78 and a standard deviation of 8? Explain.
c) In terms of percentiles, would you be better off in a class for which scores were normally distribuated with a mean of 80 and a standard deviation of 4? Explain.

a) To find the percentile corresponding to your test score, we need to calculate the z-score and then look it up in the standard normal table. The z-score formula is:

z = (x - μ) / σ

where x is the test score, μ is the mean, and σ is the standard deviation.

Plugging in the values:
x = 76
μ = 80
σ = 8

z = (76 - 80) / 8
z = -4 / 8
z = -0.5

Now, we can look up the z-score of -0.5 in the standard normal table. This gives us the percentile associated with your test score. The percentile is the percentage of scores below your score.

Using a standard normal distribution table, we find that the percentile corresponding to a z-score of -0.5 is approximately 30.85%. Therefore, your test score of 76 corresponds to the 30.85th percentile.

b) To compare percentiles, we need to calculate the z-score for the second class:

x = 76
μ = 78
σ = 8

z = (76 - 78) / 8
z = -2 / 8
z = -0.25

The z-score for the second class is -0.25. According to the standard normal table, the percentile corresponding to a z-score of -0.25 is approximately 40.39%.

Since the percentile for your test score in the second class (40.39%) is higher than the percentile in the first class (30.85%), you would be better off in a class with a mean of 78 and a standard deviation of 8.

c) To analyze the percentiles in terms of the third class, we calculate the z-score:

x = 76
μ = 80
σ = 4

z = (76 - 80) / 4
z = -4 / 4
z = -1

The z-score for the third class is -1. According to the standard normal table, the percentile corresponding to a z-score of -1 is approximately 15.87%.

Since the percentile for your test score in the third class (15.87%) is lower than the percentile in the original class (30.85%), you would be better off in a class with a mean of 80 and a standard deviation of 8 rather than a class with a mean of 80 and a standard deviation of 4.

To find the percentile corresponding to your test score, you can use the standard normal distribution table or a calculator with a normal distribution function.

a) To find the percentile for a score of 76 in a normal distribution with a mean of 80 and a standard deviation of 8, you can calculate the z-score first using the formula:

z = (x - μ) / σ

where x is your test score, μ is the mean, and σ is the standard deviation.

z = (76 - 80) / 8
z = -4 / 8
z = -0.5

Now, you can use the standard normal distribution table or a calculator to find the percentile corresponding to a z-score of -0.5. The table or calculator will give you the area to the left of -0.5, which represents the percentile. Let's assume that the percentile is P.

b) To determine if you would be better off in a class with a mean of 78 and a standard deviation of 8, you need to compare the percentile of your score in the current class with the percentile of a score of 78 in the new class.

Using the same process as in part a, calculate the z-score for a score of 78 in the new class:

z = (78 - 78) / 8
z = 0 / 8
z = 0

You can find the percentile corresponding to a z-score of 0 using the standard normal distribution table or a calculator. Let's assume it is Q.

Now, compare the percentiles P and Q. If P is greater than Q, it means your score in the current class is at a higher percentile compared to the score of 78 in the new class. In this case, you would be better off in the current class. If P is less than Q, it means your score in the current class is at a lower percentile compared to the score of 78 in the new class. In this case, you would be better off in the new class.

c) Similarly, to assess if you would be better off in a class with a mean of 80 and a standard deviation of 4, you can follow the same steps as in part b to calculate the z-score for a score of 80 in the new class and compare the respective percentiles.

Remember that the percentile indicates your position relative to other students in the class. A higher percentile means you scored better compared to your peers, while a lower percentile means you scored worse.