Question

X: Vitamin B1 content of a slice of bread μ= 33 microgram and σ= 0.005 milligram

a) According to Chebyshev’s Theorem, what is the interval of vitamin B1 content that at least 8 out of 9 slices of bread satisfy?

b) Assume that X has a normal distribution, find the exact probability of observing the interval you find in part a). Compare and discuss the results.

Answer 1

let X be the Vitamin B1 content of a slice of bread with μ= 33 microgram and σ= 0.005 milligram = 5 micrograms

a) If population distribution is not bell shaped then we use CHEBYSHEV'S THEOREM which says that at least (1 - 1/ K2) of values lie within k standard deviations of mean that is mean + k*standard deviation where k is any whole positive number greater than 1

Here, Probability = 8/9 *100 = 88.89%

So, (1- 1/ k2) = 8/9 , solving we get k = 3

which means at least 8/9 (or 88.89%) of values lie within k( equal to 3) standard deviations from mean.

so, interval is mean + k*standard deviation = 33 + 3*5 = 33 + 15 = 18 micrograms to 48 micrograms

hence, at least 8 out of 9 slices of bread contain between 18 to 48 micrograms of vitamin B1.

b) If population distribution is bell shaped (normal) then we can use standard normal distribution curve with mean = 33 and standard deviation = 5

for x = 18 , z = (18-33)/5 = - 3 and for x =48, z = (48-33)/5 = 3

So, P (18< X < 48) = P (-3 < Z < 3)

since bell curve is symmetric about mean or z = 0

hence, P (-3 < Z < 3) = P(0 < Z < 3) - P(-3 < Z < 0) = P(Z < 3) - P(Z < 0) + P(Z < 0) - P(Z < -3) = 0.999 - 0.5 + 0.5 - 0.001 = 0.998

So, 99.8% of bread slices contains vitamin B1 between 18 micrograms to 48 micrograms. which is more than what(88.89%) we got in part a

comparing part a to part b we can observe that we are more likely to find the values in between a given interval if the data or random variable is normally distributed.

note : The Chebyshev's theorem gives the minimum percentage of the data which must lie within a given number of standard deviations of the mean while the true percenatge found within the interval (regions) could be greater than what the theorem guarantees. Also the chebyshev,s theorem is applicable to all types of data diistriibution.

here is the image in case you need it.