Statistics: Analyze student exam scores
Students were given an exam with 300 multiple choice questions. The distribution of the scores were normal and the mean was 195 with a standard deviation of 30. you may find it helpful to draw out this distribution before answering the questions below
16. What were the scores of the students who were within one standard deviation of the mean?
17. What percentf the students did that include?
18. What were the scores of students who scored in the middle of the class?
19. If a's are given to students who score 90% or above, what is the minimum z-score of someone getting an a?
20. Let's say you got 235 on this test. Your colleague is in a different section of this course. Her score was 82 out of 100 on her test. in her section the mean was 72 and the standard deviation was 7. Which one of you did better? (hint: compare z-scores)
16. Z = (score-mean)/SD
Use Z = +1 and -1.
17. Mean ± 1 SD = 68.26% (Z Score is your score in terms of SD away from the mean.)
19. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.9000) and its Z score.
20. Take the hint given.
16. To find the scores of the students who were within one standard deviation of the mean, we can use the properties of a normal distribution. Since the mean score is 195 and the standard deviation is 30, we can calculate the range within one standard deviation as follows:
Lower Bound = Mean - Standard Deviation
= 195 - 30
= 165
Upper Bound = Mean + Standard Deviation
= 195 + 30
= 225
So, the scores of the students who were within one standard deviation of the mean range from 165 to 225.
17. To find the percentage of students within one standard deviation of the mean, we need to calculate the area under the normal distribution curve between the lower and upper bounds. In this case, the lower bound is 165 and the upper bound is 225.
We can calculate this using a statistical software or a normal distribution table. Assuming a symmetrical distribution, roughly 68% of the scores fall within one standard deviation of the mean.
18. To determine the scores of students who scored in the middle of the class, we need to find the range that represents the middle 50% of the scores. In other words, we need to find the range that contains the scores between the 25th and 75th percentiles.
Using the properties of the normal distribution, we can calculate the z-scores corresponding to the 25th and 75th percentiles and then convert them back to the original scores using the mean and standard deviation.
The z-score corresponding to the 25th percentile is -0.674, and the z-score corresponding to the 75th percentile is 0.674. Using these z-scores, we can calculate the corresponding scores:
Lower Bound = Mean + (Z-score * Standard Deviation)
= 195 + (-0.674 * 30)
= 175.78
Upper Bound = Mean + (Z-score * Standard Deviation)
= 195 + (0.674 * 30)
= 214.22
So, the scores of students who scored in the middle of the class range from approximately 176 to 214.
19. To find the minimum z-score for someone getting an A, we need to determine the cutoff based on the given criteria. A score of 90% or above corresponds to a z-score that includes the upper tail of the distribution.
Using a standard normal distribution table, we can find the z-score corresponding to a cumulative probability of 90%. This z-score represents the threshold for an A.
The z-score corresponding to a cumulative probability of 90% is approximately 1.28. So, the minimum z-score to get an A is 1.28.
20. To compare your score of 235 with your colleague's score of 82, we need to convert both scores to z-scores using their respective means and standard deviations.
For your score:
Z-score = (Your Score - Mean) / Standard Deviation
= (235 - 195) / 30
= 1.33
For your colleague's score:
Z-score = (Colleague's Score - Mean) / Standard Deviation
= (82 - 72) / 7
= 1.43
Comparing the z-scores, we can see that your z-score of 1.33 is lower than your colleague's z-score of 1.43. This means that you did better on the test compared to your colleague, considering the different means and standard deviations of your respective sections.