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
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
To calculate the proportion of scores falling between 20 and 40 in a normal distribution, we can use the standard normal distribution table or Z-table.
The first step is to convert the given values of 20 and 40 into Z-scores, which represent the number of standard deviations a particular value is from the mean.
To calculate the Z-score for 20:
Z = (X - μ) / σ
Z = (20 - 35) / 6
Z = -2.5
To calculate the Z-score for 40:
Z = (X - μ) / σ
Z = (40 - 35) / 6
Z = 0.83
Once we have the Z-scores, we can look up the corresponding proportions from the standard normal distribution table.
From the table, we find that the proportion of scores falling below Z = -2.5 is approximately 0.0062, and the proportion of scores falling below Z = 0.83 is approximately 0.7967.
To find the proportion of scores falling between 20 and 40, we subtract the proportion below 20 (Z = -2.5) from the proportion below 40 (Z = 0.83):
Proportion = 0.7967 - 0.0062
Proportion = 0.7905
Therefore, approximately 0.7905 or 79.05% of the scores fall between 20 and 40 in this distribution.