The scores on a sociology examination are normally distributed with a mean of 70 and a standard deviation of 10. If the instructor assigns As to 15%, Bs to 25%, Cs to 40%, Ds to 15%, and Fs to 5% of the class, find the cutoff points for grades A-D.
I started with A:
p( Z> 0.15-70/10) and got -6.99 but that doesn't make sense since there is no z score beyond -3.4 on the normal distribution table. Someone help!
A ... top 15% ... approx. 1.037 s.d. above the mean ... grade of 80
B ... next 25% ... approx. 0.253 s.d. above the mean ... grade of 73
C ... next 40% ... approx. 0.842 s.d. below the mean ... grade of 62
D ... next 15% ... approx. 1.645 s.d. below the mean ... grade of 54
To find the cutoff points for grades A, B, C, and D, we need to find the corresponding z-scores.
The z-score represents the number of standard deviations away from the mean a particular value is. To find the z-score for a specific percentile, you need to use the standard normal distribution table (also known as the z-table) or a statistical calculator.
Let's start with finding the cutoff point for grade A. The instructor assigns As to 15% of the class. Since the scores are normally distributed with a mean of 70 and a standard deviation of 10, we need to find the z-score that corresponds to the 15th percentile.
The z-score formula is: z = (x - μ) / σ
where z is the z-score, x is the value of interest (in this case, the cutoff point), μ is the mean (70), and σ is the standard deviation (10).
To find the z-score for the 15th percentile, you would look up the corresponding z-score in the z-table or use a calculator. However, since the instructor assigns As to the top 15% of the class, we need to find the z-score that corresponds to the 85th percentile (100% - 15% = 85%).
The area between the mean (μ) and the cutoff point for grade A under the standard normal distribution curve corresponds to the 85th percentile. Therefore, we need to find the z-score that corresponds to the 85th percentile.
Using the z-table, you can find that the z-score for the 85th percentile is approximately 1.04.
Now we can find the cutoff point for grade A using the formula:
cutoff point = z * σ + μ
where z is the z-score and σ is the standard deviation.
cutoff point for grade A = 1.04 * 10 + 70 = 80.4
Therefore, the cutoff point for grade A is approximately 80.4.
You can follow a similar process to find the cutoff points for grades B, C, and D by determining the corresponding percentiles, finding the z-scores, and then calculating the cutoff points using the formula mentioned above.