Scores on a statewide standardized test are normally distributed with a mean of 12.89 and a standard deviation of 1.95. Certificates are given to students whose scores are in the top 2% of those who took the test. This means that they scored better than 98% of the other test takers. Marcus received his score of 13.7 on the exam and is wondering why he didn’t receive a certificate. Show all work to determine whether Marcus’ score was high enough to earn a certificate. Write a letter to Marcus explaining whether or not he will be receiving a certificate. Include a brief summary of your statistical analysis in your letter.
I get 32% of students with grades of 13.7 or above.
try:
http://davidmlane.com/hyperstat/z_table.html
Dear Marcus,
I hope this letter finds you well. I understand that you recently received your score of 13.7 on the statewide standardized test and are curious about whether or not you will be receiving a certificate. I will be happy to assist you in analyzing your score and determining if it meets the criteria for earning a certificate.
To begin with, let's consider some statistical information about the test scores. The scores on the test are normally distributed, with a mean of 12.89 and a standard deviation of 1.95. This means that most of the scores fall near the mean of 12.89, with fewer scores being farther away from the mean.
Certificates are given to students whose scores are in the top 2% of all test takers. In other words, to earn a certificate, a student must have a score higher than 98% of all other test takers. This cutoff point for receiving a certificate is determined by calculating a z-score.
A z-score measures how many standard deviations a particular score is from the mean. To calculate the z-score for your score of 13.7, we can use the formula:
z = (x - μ) / σ
where x is the score, μ is the mean, and σ is the standard deviation.
Substituting the values, we have:
z = (13.7 - 12.89) / 1.95
z = 0.413 / 1.95
z ≈ 0.211
The next step is to find the percentile corresponding to this z-score. The percentile represents the percentage of scores that are below a given score. We can use a standard normal distribution table or a calculator to find the percentile associated with a z-score of 0.211.
Using either method, we find that a z-score of 0.211 corresponds to a percentile of approximately 58.02%. This means that your score of 13.7 is higher than approximately 58.02% of all other test takers.
Since receiving a certificate requires scoring higher than 98% of test takers, your score of 13.7 unfortunately does not meet the criteria for earning a certificate. However, it's worth mentioning that your score is still above average, considering that it is higher than approximately 58.02% of test takers.
I hope this analysis has provided you with a clear understanding of why you did not receive a certificate. Remember, the certificate is awarded to only the top 2% of test takers, which is an exceptional accomplishment. Your score of 13.7 is still commendable, and you should be proud of your performance.
If you have any further questions or need additional clarification, please feel free to ask. Best of luck in your future endeavors.
Warm regards,
Explain Bot