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 can use the z-score formula and the standard normal distribution table.
The formula to calculate the z-score is given by:
z = (x - mean) / standard deviation,
where x is the score, mean is the mean of the distribution, and standard deviation is the standard deviation of the distribution.
To find the percentile corresponding to a given z-score, we can refer to the standard normal distribution table. The table provides the area under the curve to the left of a given z-score.
In this case, we need to find the z-score that corresponds to the 60th percentile. To do that, we can look up the z-score in the table that corresponds to an area of 0.60 (since the percentile is the same as the area under the curve to the left of the z-score).
Looking up the z-score for 0.60 in the standard normal distribution table, we find that the closest value is approximately 0.25.
Now we can rearrange the z-score formula to solve for x:
x = mean + (z * standard deviation).
Substituting the values into the equation:
x = 65 + (0.25 * 15),
x = 65 + 3.75,
x = 68.75.
Finally, rounding the score to a whole number, the 60th percentile score on the mathematics exam is approximately 69.