Scores on a mathematics examination appear to follow a normal distribution with mean of 65 and standard deviation of 15. The instructor wishes to give a grade of “C” to students scoring between the 60th and 70th percentiles on the exam.
What score represents the 60th percentile score on the mathematics exam? Place your answer in the blank, rounded to a whole number. For example, 62 would be a legitimate entry.
To find the score that represents the 60th percentile on the mathematics exam, we need to use the standard normal distribution table or a statistical calculator.
Here's how to do it using the standard normal distribution table:
1. Start by standardizing the score using the formula:
z = (x - mean) / standard deviation
In this case, the mean (μ) is 65 and the standard deviation (σ) is 15. So, the formula becomes:
z = (x - 65) / 15
2. Next, find the z-score that corresponds to the 60th percentile. The 60th percentile is equivalent to a cumulative probability of 0.60. Using the standard normal distribution table, look for the z-value that is closest to 0.60. Let's call this z-value z_60.
3. Once you find the z-value, substitute it back into the formula and solve for x:
z_60 = (x - 65) / 15
Rearrange the formula to solve for x:
x = (z_60 * 15) + 65
Calculate x using the z-value from the table, then round it to the nearest whole number, and that will give you the score representing the 60th percentile.
Please note that the standard normal distribution table provides the cumulative probability up to a certain z-value, so you may need to do some interpolation if the exact value is not listed in the table.