The mean grade in this class last semester was 78.3, and the variance was 49 . The distribution of grades was unimodal and symmetrical. Using this information determine the probability that “Joe”, a random student you know nothing about, other than the fact that they are taking this class next semester will receive a grade that is:

Above 90
Between 85 and 95
Between 80 and 90
Lower than 65
Lower than 65 OR above 90

My question is how do you approach this question would i have to do something along these lines?

90-78.3/49?

Z = (score-mean)/SD

SD = √variance

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.

Either-or probabilities are found by adding the individual probabilities.

To find the probability of Joe receiving a certain grade, we need to use the properties of a normal distribution. Since we know that the distribution of grades is unimodal and symmetrical, we can assume it follows a normal distribution.

To approach this problem, we need to standardize the scores using z-scores and then use the standard normal distribution table (also known as z-table) to find the probabilities.

To standardize a score, we use the formula:
z = (x - μ) / σ
Where:
- x is the individual score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

Now, let's work through each probability:

1. Probability of getting a grade above 90:
First, calculate the z-score:
z = (90 - 78.3) / √49
Then, using the z-table, find the area to the right of the z-score.

2. Probability of getting a grade between 85 and 95:
Calculate the z-scores for both values:
z1 = (85 - 78.3) / √49
z2 = (95 - 78.3) / √49
Find the area between these two z-scores using the z-table.

3. Probability of getting a grade between 80 and 90:
Calculate the z-scores for both values:
z1 = (80 - 78.3) / √49
z2 = (90 - 78.3) / √49
Find the area between these two z-scores using the z-table.

4. Probability of getting a grade lower than 65:
Calculate the z-score:
z = (65 - 78.3) / √49
Find the area to the left of this z-score using the z-table.

5. Probability of getting a grade lower than 65 or above 90:
Since these two events are mutually exclusive, you can calculate their individual probabilities and add them together.

Remember to consult the standard normal distribution table (z-table) to find the probabilities associated with the calculated z-scores.