Large group of students took a test in English and the final grades have a mean of 75 and a standard deviation of 8. If we can approximate the distribution of these grades by a normal distribution. What percentage of the students
(ii) Should pass the test (grades >= 60)
(iii) Should fail the test(grades <60?
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
related to the Z score.
An urn contains7 white balls and 3 yellow balls. If Tim chooses 8 balls at random from the urn, what is the probability that he will select6 white balls and 2 yellow balls? Round your answer to decimal places.
To find the percentage of students who should pass the test (grades >= 60) and the percentage of students who should fail the test (grades < 60), we need to use the properties of the normal distribution.
Since we know the mean (μ = 75) and the standard deviation (σ = 8), we can use these values to standardize the scores and then use the standard normal distribution table (also known as the Z-table) to find the corresponding percentages.
(i) To find the percentage of students who should pass the test (grades >= 60), we need to find the area under the normal curve to the right of the threshold value (60) when the scores are standardized. First, we calculate the Z-score:
Z = (X - μ) / σ
Here, X is the threshold value (60), μ is the mean (75), and σ is the standard deviation (8).
Z = (60 - 75) / 8 = -1.875
Next, we use the Z-table to find the area to the right of Z = -1.875. The Z-table provides the percentage of values that are less than a given Z-score. To find the percentage of values greater than Z = -1.875, we subtract the corresponding area from 1 (since the total area under the curve is 1).
Using the Z-table, we find that the area to the right of Z = -1.875 is approximately 0.9699. Therefore, the percentage of students who should pass the test (grades >= 60) is approximately:
Percentage = (1 - 0.9699) * 100 ≈ 3.01%
(ii) To find the percentage of students who should fail the test (grades < 60), we can subtract the percentage of students who should pass (grades >= 60) from 100%.
Percentage = 100% - 3.01% = 96.99%
Therefore, approximately 96.99% of the students should fail the test (grades < 60).