Approximately 16% of undergraduates at the University have a writing score above 68 (on a 100 point scale). Note: writing scores are normally distributed. if a standard deviation is 8 what is the mean writting score
To find the mean writing score, we will use the information given about the percentage of students with scores above a certain value.
1. Start by identifying the z-score associated with the given percentage. The z-score represents the number of standard deviations the score is from the mean. We can use a standard normal distribution table or calculator to find this.
From the given information, we know that 16% of students have scores above 68. This means that 100% - 16% = 84% of students have scores below or equal to 68. The corresponding z-score can be found using a standard normal distribution table or calculator. In this case, a z-score of approximately 0.9945 corresponds to the 84th percentile.
2. Use the z-score formula to find the mean. The z-score formula is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the given value (68 in this case)
- μ is the mean
- σ is the standard deviation
Rearranging the formula to solve for the mean, we have:
μ = x - (z * σ)
Plugging in the given values:
μ = 68 - (0.9945 * 8)
μ ≈ 68 - 7.956
μ ≈ 60.044
Therefore, the mean writing score is approximately 60.044.