A set of data is normally distributed with a mean of 1000 and standard deviation of 100.
· What would be the standard score for a score of 1100?
· What percentage of scores is between 1000 and 1100?
· What would be the percentile rank for a score of 1100?
A. z(1100)= (1100-1000)/100= 100/100= 1
B. p(1100<=x<=1000) = (1<=z<=0) = 0.3486 { I am not certain on this part of the question}
C. If the last question is correct. It should be the 34th percentile.
1. Yes.
2. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between mean and Z = +1. (You we close.)
3. No, the percentile rank is the proportion of scores equal to or below a particular score. Notice that your score is above the mean.
To find the answers to these questions, you need to understand the concept of standard scores (also known as z-scores) and the properties of the normal distribution.
1. Standard score (z-score) for a score of 1100:
The standard score measures how many standard deviations a data point is away from the mean. The formula to calculate the standard score is: z = (X - μ) / σ, where X is the individual score, μ is the mean, and σ is the standard deviation.
So, in this case, the standard score (z) for a score of 1100 would be: z = (1100 - 1000) / 100 = 100 / 100 = 1.
2. Percentage of scores between 1000 and 1100:
To find out what percentage of scores lie between two values in a normal distribution, you can use the z-score and the standard normal distribution table (also known as the Z-table). The Z-table provides the area under the curve for different z-scores.
To find the percentage of scores between 1000 and 1100:
- Convert both scores to their respective z-scores: For 1000, z = (1000 - 1000) / 100 = 0, and for 1100, z = (1100 - 1000) / 100 = 1.
- Look up these z-scores in the Z-table to find their respective area under the curve.
- Calculate the difference between the two areas to find the percentage.
In this case, I cannot calculate it precisely without the Z-table or the specific software that can calculate these values, but you can refer to the Z-table or use statistical software like Excel, SPSS, or Python to find the percentage.
3. Percentile rank for a score of 1100:
The percentile rank represents the percentage of scores that are equal to or below a certain score. To find the percentile rank for a specific score, you need to find the area under the normal curve up to that score.
Using the same z-score of 1 for a score of 1100, you can again use the Z-table to find the area under the curve up to a z-score of 1. The Z-table will give you the corresponding area, which can then be converted to a percentile rank. In this case, it would be around the 34th percentile, assuming the calculations are accurate.
Remember, the Z-table or statistical software is needed to make accurate calculations for the exact proportions or percentile ranks.