Respuesta :
Answer:
The answer should be 99.85%, but no option is close enough. Please check if there is any mistake at the question.
Step-by-step explanation:
Empirical Rule of Normal Distribution:
It states that for a Normal Distribution (bell-shaped), the percentages of the data will follow this rule:
- About 68% of the data lies within 1 standard deviation of the mean (-1 < Z < 1)
- About 95% of the data lies within 2 standard deviations of the mean (-2 < Z < 2)
- About 99.7% of the data lies within 3 standard deviations of the mean (-3 < Z < 3)
Given:
- mean (μ) = 103
- standard deviation (σ) = 15
- x = 58
By using the Z-Score Formula, we can find x lies in which standard deviation:
[tex]\boxed{Z=\frac{x-\mu}{\sigma} }[/tex]
[tex]\displaystyle Z=\frac{58-103}{15}[/tex]
[tex]Z=-3[/tex]
Probability that IQ scores that are at least 58 → means Probability of Z > -3, which is the green area of the picture. To calculate using the Empirical Rule:
P(Z > -3) = P(-3 < Z < 3) + P(Z > 3)
= 99.7% + [(100% - 99.7%) ÷ 2]
= 99.7% + 0.15%
= 99.85%
Final answer:
Using the empirical rule to analyze a normal distribution of IQ scores with a mean of 103 and a standard deviation of 15, we find that approximately 99.85% of scores are at least 58. The closest answer choice, though considerably lower, is 39%.
Explanation:
To determine what percentage of IQ scores are at least 58, we can apply the empirical (68-95-99.7) rule to the normal distribution of IQ scores. Given a mean IQ score of 103 and a standard deviation of 15, we want to find the percentage of scores falling to the left of 58.
First, we calculate how many standard deviations below the mean an IQ score of 58 is:
(58 - 103) / 15 = -45 / 15 = -3.
An IQ score of 58 is three standard deviations below the mean. According to the empirical rule:
- 68% of the data falls within one standard deviation of the mean.
- 95% of the data falls within two standard deviations of the mean.
- 99.7% of the data falls within three standard deviations of the mean.
Therefore, the percentage of people with an IQ score higher than 58 would be approximately 100% minus the percentage accounted for in the smallest half of the third standard deviation (which is 0.15% from empirical rule).
100% - 0.15% = 99.85%.
To answer the original question, since options a, b, c, and d do not include 99.85%, there might be a mistake in the question or answer choices. However, if we had to choose the closest option, it would be d. 39%, as all the other choices are less than 39%, and we've identified that a much larger percentage of scores are at least 58.
