a mean score on a standardized test is 50 with a standard deviation of 10. What are the following:
1. what scores fall between -1 and +1 standard deviation?
2. what percent of all scores fall between -1 and +1 standard deviation?
3. what score falls at +2 standard deviations?
4. What percentage of scores falls between +1 and +2 standard deviations?
Z score is your score in terms of standard deviations.
Z = (score-mean)/SD
For the percentages, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Multiply by 100 for the percentage.
1. To find the scores that fall between -1 and +1 standard deviation from the mean, we need to calculate the lower and upper bounds.
Lower bound = mean - 1 * standard deviation
Lower bound = 50 - 1 * 10
Lower bound = 40
Upper bound = mean + 1 * standard deviation
Upper bound = 50 + 1 * 10
Upper bound = 60
So, the scores that fall between -1 and +1 standard deviation are 40 to 60.
2. To find the percentage of all scores that fall between -1 and +1 standard deviation, we need to calculate the probability using the standard normal distribution.
The area under the normal curve between -1 and +1 standard deviations from the mean is approximately 68%. This means that 68% of all scores will fall within this range.
3. To find the score that falls at +2 standard deviations from the mean, we need to calculate the value.
Score at +2 standard deviations = mean + 2 * standard deviation
Score at +2 standard deviations = 50 + 2 * 10
Score at +2 standard deviations = 70
So, the score that falls at +2 standard deviations is 70.
4. To find the percentage of scores that fall between +1 and +2 standard deviations, we need to calculate the probability using the standard normal distribution.
The area under the normal curve between +1 and +2 standard deviations from the mean is approximately 13.6%. This means that approximately 13.6% of all scores will fall within this range.
To answer these questions, we will use a standard normal distribution (also known as a Z-distribution), which is based on the mean score and standard deviation provided.
1. To determine the scores that fall between -1 and +1 standard deviations, we need to convert these values into Z-scores. The formula to calculate the Z-score is: Z = (X - mean) / standard deviation.
For the lower limit: Z = (-1 - 50) / 10 = -51 / 10 = -5.1
For the upper limit: Z = (1 - 50) / 10 = -49 / 10 = -4.9
Next, we consult the standard normal distribution table or use a calculator to find the corresponding area under the curve between these two Z-scores. The table would show that the area is approximately 0.6826.
2. To find the percentage of all scores that fall between -1 and +1 standard deviations, we can multiply the proportion (0.6826) by 100. So, approximately 68.26% of all scores fall between -1 and +1 standard deviations.
3. To find the score that falls at +2 standard deviations, we can use the Z-score formula. Z = (X - mean) / standard deviation. Since the mean is 50 and the standard deviation is 10, we can rearrange the formula to solve for X.
For +2 standard deviations: 2 = (X - 50) / 10
Solving the equation, we get: X - 50 = 20
Adding 50 to both sides, we find: X = 70
Therefore, the score at +2 standard deviations is 70.
4. To find the percentage of scores that fall between +1 and +2 standard deviations, we can follow the same process as in question 1. First, convert the values to Z-scores:
For the lower limit: Z = (1 - 50) / 10 = -49 / 10 = -4.9
For the upper limit: Z = (2 - 50) / 10 = -48 / 10 = -4.8
Next, consult the standard normal distribution table or use a calculator to find the area under the curve between these two Z-scores. The table would show that the area is approximately 0.1359.
Multiply the proportion (0.1359) by 100 to find the percentage. So, approximately 13.59% of scores fall between +1 and +2 standard deviations.