Respuesta :
Answer:
a) 99.7% of the organs will be between 250 and 420 grams.
b) 95% of the organs weighs between 280 grams and 400 grams.
c) 5% of the organs weighs less than 280 grams or more than 400 grams.
d) 97.5% of the organs weighs between 250 grams and 400 grams.
Step-by-step explanation:
The mean value of the weight of an organ is 340 grams and the standard deviation is 30 grams, so [tex]\mu = 340, \sigma = 30[/tex].
(a) About 99.7% of organs will be between what weights?
The Empirical Rule states that 99.7% of the values of X of a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] belong to the following interval
[tex]\mu - 3\sigma \leq X \leq \mu + 3\sigma[/tex]
We have that [tex]\mu = 340, \sigma = 30[/tex], so:
[tex]\mu - 3\sigma \leq X \leq \mu + 3\sigma[/tex]
[tex]340 - 3(30) \leq X \leq 340 + 3(30)[/tex]
[tex]250 \leq X \leq 420[/tex]
99.7% of the organs will be between 250 and 420 grams.
(b) What percentage of organs weighs between 280 grams and 400 grams?
280 grams is 2 standard deviations below the mean.
400 grams is 2 standard deviations above the mean.
The Empirical rule states that 95% of the weights of the organs are between 2 standard deviations of the mean, so
95% of the organs weighs between 280 grams and 400 grams.
(c) What percentage of organs weighs less than 280 grams or more than 400 grams?
95% of the organs weighs between 280 grams and 400 grams. This means that 5% of the organs weighs less than 280 grams or more than 400 grams.
(d) What percentage of organs weighs between 250 grams and 400 grams?
250 grams is 3 standard deviations below the mean.
The Empirical rule states that 50% of the weights are above the mean, and 50% are below. Of those that are below, 100% weights at least 3 standard deviations below the mean.
400 grams is 2 standard deviations above the mean.
Of the values that are above the mean, 95% of them are at most 2 standard deviations above the mean.
So:
[tex]P = 0.5 + 0.95(0.5) = 0.5 + 0.475 = 0.975[/tex]
97.5% of the organs weighs between 250 grams and 400 grams.
You can use the empirical rules for probability related to normal distribution to get the needed percentage.
The answers are
- About 99.7% of organs will be between 250 and 430
- 95% organs weighs in between 280 grams and 400 grams
- 5% percentage of organs weighs less than 280 grams or more than 400 grams
- Percentage of organs weighs between 250 grams and 400 grams is 97.35% approximately.
What is empirical rule for normal distribution?
According to the empirical rule, also known as 68-95-99.7 rule, the percentage of values that lie within an interval with 68%, 95% and 99.7% of the values lies within one, two or three standard deviations of the mean of the distribution.
We can write it symbolically as:
[tex]P(\mu - \sigma < X < \mu + \sigma) = 68\%\\P(\mu - 2\sigma < X < \mu + 2\sigma) = 95\%\\P(\mu - 3\sigma < X < \mu + 3\sigma) = 99.7\%[/tex]
where we had [tex]X \sim N(\mu, \sigma)[/tex]
where mean of distribution of X is [tex]\mu[/tex] and standard deviation from mean of distribution of X is [tex]\sigma[/tex]
Using the above rule to find the needed percentage
Let the weight of organs be measured as values of random variable X
Then we have, [tex]X \sim N(\mu = 340, \sigma = 30)[/tex]
Thus,
a) For 99,7%, we have the interval of weights as [tex][\mu - 3\sigma, \mu + 3\sigma][/tex]
Evaluating, we get [tex][\mu - 3\sigma, \mu + 3\sigma] = [250, 430][/tex]
b) Since 280 is 340 - 60 which is 340 - twice of 30, and we 400 is 340 + 60 = 340 - twice of 30,
thus,
[tex]P(\mu - 2\sigma < X < \mu + 2\sigma) = 95\%\\\\P(280 < X < 400) = 95\%[/tex]
Thus, 95% organs weighs in between 280 grams and 400 grams.
c) We can use the complement of the event.
Thus, if we have [tex]P(280 < X < 400) = 95\%[/tex]
Then we have [tex]P(X < 280) + P(X > 400) = 100 - 95\% = 5\%[/tex]
d) Since we know that normal distribution is symmetric about its mean, thus,
[tex]P(X < \mu - k) = P(X > \mu + k)[/tex]
where k is a constant.
Putting k = 2 sigma, and using the fact that [tex]P(280 < X < 400) = 95\%[/tex]
[tex]P(X < 280) + P(X > 400) = 100 - 95\% = 5\%\\2P(X>400) = 5\%\\P(X > 400) = 2.5\%\\P(X\leq 400) = 100 - 2.5\% = 97.5\%[/tex]
Similarly putting k = 3, we get:
[tex]P(250 < X < 430) = 99.7\%\\P(X<250) + P(X>430) = 0.3\%\\2P(X<250) = 0.3\%\\P(X<250) = 0.15\%[/tex]
Thus,
probability that weights of organs is between 250 and 400 grams is
[tex]P(250 < X < 400) = P(X<400) - P(X < 250) = 97.5\% - 0.15\% = 97.35\%[/tex]
Thus, percentage of organs weighs between 250 grams and 400 grams is 97.35% approximately.
Thus,
The answers are
- About 99.7% of organs will be between 250 and 430
- 95% organs weighs in between 280 grams and 400 grams
- 5% percentage of organs weighs less than 280 grams or more than 400 grams
- Percentage of organs weighs between 250 grams and 400 grams is 97.35% approximately.
Learn more about empirical rule here:
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