if there are 145 test questions, and the mean of the test scores for a class is 100 with a standard devation of 15, what percentage of the people taking the test would have the following scores: scored b/w 100 &115, scored b/w 100 & 115,scored b/w 0 & 100, scored b/w 0 & 145, or scored b/w 0 &85?
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 find the percentage of people who scored within a specific range, you can use the concept of the normal distribution. In this case, you have the mean (100) and the standard deviation (15) of the test scores. The normal distribution is often represented by a bell curve.
To calculate the percentage, you will use the cumulative distribution function (CDF) of the normal distribution. The CDF provides the probability that a randomly selected value from the distribution falls below a certain value (in this case, the test score). By subtracting the CDF values, you can find the percentage of people who fall within a specific range.
Let's calculate the percentages for the given ranges:
1. Scored between 100 and 115:
To find the percentage of people who scored between two values, you need to subtract the CDF of the lower value from the CDF of the upper value. First, standardize the values using the formula: z = (x - mean) / standard deviation.
For the lower bound (x = 100):
z1 = (100 - 100) / 15 = 0
For the upper bound (x = 115):
z2 = (115 - 100) / 15 = 1
Now, use a standard normal distribution table or a statistical calculator to find the CDF for these standardized values.
CDF1 = CDF(z1) = CDF(0)
CDF2 = CDF(z2) = CDF(1)
Finally, calculate the percentage: Percentage = (CDF2 - CDF1) * 100
2. Scored between 0 and 100:
This range includes all values less than or equal to 100, so we only need the CDF for the lower bound (x = 100).
z = (100 - 100) / 15 = 0
Percentage = CDF(z) * 100
3. Scored between 0 and 145:
This range includes all possible values, so the percentage is 100%.
4. Scored between 0 and 85:
Similar to the previous range, this range includes all values less than or equal to 85.
z = (85 - 100) / 15 = -1
Percentage = CDF(z) * 100
You can now use the calculated values along with a standard normal distribution table or a statistical calculator to find the respective percentages.