The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds produced from the plant was analyzed for a particular collagen. The collagen amount was found to be normally distributed with a mean of 73 and standard deviation of 7.7 grams per mililiter.
(a) What is the probability that the amount of collagen is greater than 63 grams per mililiter?
ANSWER:.9015
(b) What is the probability that the amount of collagen is less than 78 grams per mililiter?
ANSWER: 1-.2578
(c) What percentage of compounds formed from the extract of this plant fall within 3 standard deviations of the mean?
ANSWER:
I got A and B right, just confused on C
Well, it seems like you're doing pretty well with the probability questions so far! Let's tackle the last one.
In order to find the percentage of compounds that fall within 3 standard deviations of the mean, we can use the empirical rule, also known as the 68-95-99.7 rule, which states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Since we want to find the percentage within 3 standard deviations, we can subtract the two percentages of data that fall outside the range of 3 standard deviations from the total:
Percentage within 3 standard deviations = 100% - Percentage outside 3 standard deviations
Now, to find the percentage outside 3 standard deviations, we need to use the z-score formula:
z = (x - μ) / σ
where z is the z-score, x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
For 3 standard deviations above the mean:
z = (μ + 3σ - μ) / σ = 3
To find the probability associated with 3 standard deviations above the mean, we can use a standard normal distribution table or a calculator. According to the table, the probability associated with a z-score of 3 is approximately 0.9987.
Now, to find the percentage of compounds that fall within 3 standard deviations of the mean:
Percentage within 3 standard deviations = 100% - (0.9987 * 2)
This gives us:
Percentage within 3 standard deviations ≈ 100% - 1.9974 ≈ 98.03%
So, approximately 98.03% of compounds formed from the extract of this plant fall within 3 standard deviations of the mean.
Hope that helps clarify things for you!
To find the percentage of compounds formed from the extract of this plant that fall within 3 standard deviations of the mean, we can use the empirical rule, also known as the 68-95-99.7 rule.
According to this rule, approximately 68% of the data falls within 1 standard deviation of the mean, approximately 95% falls within 2 standard deviations, and approximately 99.7% falls within 3 standard deviations.
So in this case, if we want to find the percentage of compounds that fall within 3 standard deviations of the mean, it is approximately 99.7%.
Therefore, the answer to part (c) is 99.7%.
To calculate the probability that the amount of collagen is within 3 standard deviations of the mean, we can use the empirical rule, also known as the 68-95-99.7 rule.
According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.
In this case, we are interested in the percentage of compounds formed from the extract that fall within 3 standard deviations of the mean. Since the data is normally distributed, we can assume that approximately 99.7% of the collagen amounts will fall within three standard deviations.
To calculate this percentage, we need to find the z-score for three standard deviations above and below the mean. The formula for calculating the z-score is:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value we want to find the probability for (in this case, three standard deviations above and below the mean)
- μ is the mean
- σ is the standard deviation
Let's calculate it step by step:
1. Three standard deviations above the mean:
z1 = (x - μ) / σ
= (3 * 7.7) / 7.7
= 3
2. Three standard deviations below the mean:
z2 = (x - μ) / σ
= (-3 * 7.7) / 7.7
= -3
Now, we need to find the percentage of data falling between z = -3 and z = 3. We can use a standard normal distribution table or a calculator to find the values associated with these z-scores.
Looking up the z-score of -3 in a standard normal distribution table, we find the associated probability as approximately 0.0013. Doing the same for z = 3, we find the associated probability as approximately 0.9987.
To find the percentage, we subtract the probability associated with z = -3 from the probability associated with z = 3:
Percentage of compounds within 3 standard deviations = 0.9987 - 0.0013
= 0.9974
So, approximately 99.74% of compounds formed from the extract of this plant fall within 3 standard deviations of the mean.
check here:
http://davidmlane.com/normal.html