"suppose that IQ scores have a bell-shaped distribution with a mean of 97 and standard deviation of 17.Using empirical rule, what percentage of IQ score are less than 46?"
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To determine the percentage of IQ scores that are less than 46 using the empirical rule, we can utilize the concept of standard deviations.
The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean
- Approximately 95% falls within 2 standard deviations of the mean
- Approximately 99.7% falls within 3 standard deviations of the mean
Given that the mean IQ score is 97 and the standard deviation is 17, we can calculate the number of standard deviations that 46 is below the mean.
Z-score can be calculated using the formula:
Z = (X - μ) / σ
Where:
- Z represents the number of standard deviations below or above the mean
- X is the value we want to calculate the Z-score for (in this case, 46)
- μ is the mean (97)
- σ is the standard deviation (17)
Substituting the values:
Z = (46 - 97) / 17
Calculating:
Z = -3
Since 46 is 3 standard deviations below the mean, we can use the empirical rule to estimate the percentage of IQ scores that are less than 46. According to the empirical rule, approximately 99.7% (or nearly all) of the data falls within 3 standard deviations of the mean.
Therefore, the percentage of IQ scores less than 46 is approximately 99.7%.
To calculate the percentage of IQ scores that are less than 46 using the empirical rule, we need to determine how many standard deviations away from the mean 46 is and then refer to the empirical rule to find the corresponding percentage.
1. Calculate the z-score:
The z-score measures how many standard deviations a particular value is away from the mean. It is calculated using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
In this case, for an IQ score of 46:
z = (46 - 97) / 17
2. Calculate the percentage using the empirical rule:
The empirical rule states that for a bell-shaped distribution, approximately:
- 68% of values will fall within one standard deviation of the mean.
- 95% of values will fall within two standard deviations of the mean.
- 99.7% of values will fall within three standard deviations of the mean.
Since we want to know the percentage of scores less than 46, we need to calculate the area under the distribution curve to the left of the z-score we calculated.
Using a z-score table or statistical software, we can find the percentage corresponding to the z-score. Let's assume it is P(Z < z).
So, the percentage of IQ scores less than 46 will be P(Z < z) which corresponds to the area under the curve to the left of the z-score.
Remember to calculate z correctly, and use a z-table to look up the corresponding percentage.
Please note that the exact percentage may vary depending on the level of precision used when looking up z-scores in the table or using statistical software.