A mean score on a standardized test is 50 with a standard deviation of 10. Answer the following
a. What scores fall between –1 and +1 standard deviation?
b. What percent of all scores fall between –1 and +1 standard deviation?
c. What score falls at +2 standard deviations?
d. What percentage of scores falls between +1 and +2 standard deviations?
IF THERE IS A FORMULA CAN YOU PROVIDE IT.
Z = (score-mean)/SD
Z = raw scorre interms of standard deviations.
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. Insert into equation above.
What percentage of scores falls between +1 and +2 standard deviations?
To answer these questions, we typically use the concept of the z-score. The z-score measures how many standard deviations a particular score is above or below the mean.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the given score
- μ is the mean
- σ is the standard deviation
Let's proceed to answer each question using this formula.
a. To find the scores that fall between -1 and +1 standard deviation, we need to calculate the z-scores for both values and see which scores correspond to these z-scores.
For -1 standard deviation:
z1 = (-1 * 10) + 50 = 40
For +1 standard deviation:
z2 = (1 * 10) + 50 = 60
So, the scores that fall between -1 and +1 standard deviation are from 40 to 60.
b. To calculate the percentage of scores that fall between -1 and +1 standard deviation, we need to find the proportion of the normal distribution that lies between z1 and z2.
Using a standard normal distribution table or a calculator, we can find the probability associated with these z-scores. The value represents the proportion of scores falling within that range.
c. To find the score that falls at +2 standard deviations, we need to use the formula again. Let's substitute the values:
z3 = (2 * 10) + 50 = 70
So, the score that falls at +2 standard deviations is 70.
d. To calculate the percentage of scores that fall between +1 and +2 standard deviations, we need to calculate the proportion of the distribution that lies between z2 and z3.
Using a standard normal distribution table or calculator, we can find the probability associated with these z-scores.
Note: The percentages and probabilities mentioned in b) and d) can be obtained from standard normal distribution tables or by using a calculator or software with normal distribution functions.