point) IQ scores have mean of 100 and a standard deviation of 15. Mary has an IQ of 103. What is the difference between Mary’s IQ and the mean? Answer: () Convert Mary’s IQ score to a z-score_ Answer:
The Correct Answer and Explanation is:
To answer your question, let’s break it down step by step.
1. Difference Between Mary’s IQ and the Mean
We are given the following information:
- The population mean IQ is 100.
- The standard deviation is 15.
- Mary’s IQ is 103.
The difference between Mary’s IQ and the mean can be calculated using this simple formula: Difference=Mary’s IQ−Mean IQ\text{Difference} = \text{Mary’s IQ} – \text{Mean IQ}Difference=Mary’s IQ−Mean IQ
Substitute the given values: Difference=103−100=3\text{Difference} = 103 – 100 = 3Difference=103−100=3
So, the difference between Mary’s IQ and the mean is 3.
2. Converting Mary’s IQ to a Z-Score
To convert Mary’s IQ to a z-score, we use the formula for the z-score: Z=X−μσZ = \frac{X – \mu}{\sigma}Z=σX−μ
Where:
- XXX is the raw score (Mary’s IQ = 103),
- μ\muμ is the mean (100),
- σ\sigmaσ is the standard deviation (15).
Substitute the given values: Z=103−10015=315=0.2Z = \frac{103 – 100}{15} = \frac{3}{15} = 0.2Z=15103−100=153=0.2
So, Mary’s z-score is 0.2.
Explanation:
- Difference between IQ and the mean: This simply tells you how much higher or lower a specific IQ score is compared to the average (mean). In this case, Mary’s IQ is 3 points above the mean.
- Z-Score: The z-score is a measure of how many standard deviations a particular score is from the mean. A z-score of 0.2 means Mary’s IQ is 0.2 standard deviations above the average IQ of the population. Z-scores are useful for comparing different data points from different distributions, as they standardize the values.
In summary:
- The difference between Mary’s IQ and the mean is 3.
- Mary’s z-score is 0.2.
