SOLUTION: For a certain species of butterfly, the length of a wing (at its longest point) is normally distributed with a mean of 1 cm and a standard deviation of 0.2 cm.
Which is MORE LIK
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Question 1183363: For a certain species of butterfly, the length of a wing (at its longest point) is normally distributed with a mean of 1 cm and a standard deviation of 0.2 cm.
Which is MORE LIKELY: a.) a single butterfly has wings that are shorter than 0.8 cm, or b.) a sample of 50 butterflies has an average wing length that is shorter than 0.8 cm.
Explain your reasoning.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
with a sample of one butterfly, you use the standard deviation.
the population mean is 1 cm with a standard deviation of .2 cm.
the single butterfly has wings that are shorter than .8 cm.
the z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard deviation when you are dealing with a sample of one.
s is the standard error when you are dealing with the mean of a sample of size greater than 1.
for the sample of 1, the formula becomes z = (.8 - 1) / .2 = -1.
the probability of getting a z-score less than -1 would be equal to .1586552596.
with a sample of 50 butterflies that has an average wing length that is shorter than .8 cm, the z-score becomes:
z = (.8 - 1) / s
s is the standard error.
the standard error is equal to the standard deviation divided by the square root of the sample size.
s = .2 / sqrt(50) = .02828
formula becomes z = (.8 - 1) / .02828 = -7.072135785.
the mean of the sample being less than .8 is decidedly LESS likely than the length of the wings of one butterfly being less than .8.
the mean of the sample of 50 contains measurements of 50 butterflies, rather than just one.
the standard error, which is the standard deviation of the distribution of sample means, will become smaller when the sample size is larger.
that's why the standard error formula is equal to the standard deviation divided by the square root of the sample size.
with the standard error being less than the standard deviation, the same value of x will have a higher z-score which means the probability of getting a score less than that will be less.
-.2 / .02828 give you a higher z-score than -.2 / .2
a higher z-score means less probability of getting a score less than .8.
the one with the lower z-score is more probable.
that's the measurement from the single element rather than the mean of the sample of 50 elements.
you will see more variability in the length of the wings from a sample of one butterfly than from the mean of a sample of 50 butterflies.
because of the greater variability, the probability that you will find a butterfly with wings less than .8 cm is greater than what you will find with the mean of a sample of 50 butterflies.
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