the scores on a 100 pt test are normally distributed with a mean of 80 and a standard deviation of 6. a student's score places him between the 69th and 70th percentile. what is his score?
To find the student's score that places him between the 69th and 70th percentile, we can use the concept of z-scores.
A z-score measures the number of standard deviations a data point is from the mean of a distribution. It helps us standardize scores so that we can compare them across different distributions.
In this case, we know that the mean is 80 and the standard deviation is 6. To find the z-score corresponding to the desired percentiles, we can use a Z-score table or a calculator.
The z-score corresponding to the 69th percentile is approximately -0.439 (or -0.44 if rounding to two decimal places), and the z-score corresponding to the 70th percentile is approximately -0.374 (or -0.37 if rounding to two decimal places).
To calculate the student's score, we can use the formula for z-scores:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the data point (student's score)
- μ is the population mean (80)
- σ is the standard deviation (6)
Rearranging the formula, we can solve for x:
x = z * σ + μ
Calculating the student's score:
For the z-score of -0.439:
x1 = -0.439 * 6 + 80 ≈ 76.64 (rounded to two decimal places)
For the z-score of -0.374:
x2 = -0.374 * 6 + 80 ≈ 76.75 (rounded to two decimal places)
Therefore, the student's score that places him between the 69th and 70th percentiles is approximately 76.64 to 76.75 (rounded to two decimal places).