A distribution of 800 test scores in a biology course was approximately normally distributed with a mean of 35 and a standard deviation of 6. Calculate the proportion of scores falling between 20 and 40
To calculate the proportion of scores falling between 20 and 40, we need to use the standard normal distribution and the z-score formula.
The z-score formula is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the raw score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
First, we need to calculate the z-score for the lower bound (20) and the upper bound (40).
For the lower bound (20):
z1 = (20 - 35) / 6
z1 = -15 / 6
z1 ≈ -2.5
For the upper bound (40):
z2 = (40 - 35) / 6
z2 = 5 / 6
z2 ≈ 0.83
Now, we can use the standard normal distribution table or a calculator to find the proportion of scores falling between these two z-scores.
Looking up the z-scores in the standard normal distribution table, we find the following values:
For z = -2.5, the proportion is approximately 0.0062.
For z = 0.83, the proportion is approximately 0.7967.
To find the proportion of scores falling between these two z-scores, we subtract the lower value from the upper value:
Proportion = 0.7967 - 0.0062
Proportion ≈ 0.7905
Therefore, approximately 79.05% of the scores fall between 20 and 40 in this distribution.