Question

Diameters of Lemon grown on citrus farm are approximately normally distributed with mean 7cm. it is known that approximately 99.7% of lemons have diameters between 5.5cm and 8.5cm. Using empirical rule,calculate approximately what percentage of lemons have diameters between 6cm and 7.5cm

Answer 1

Answer

By empirical rule, we know that 99.7% of data lie within 3 standard deviations from the mean. We have mean = 7

So, we will calculate value of standard deviation

Lower limit= mean - 3(sd)

we have lower limit = 5.5 and mean = 7, sd is standard deviation

setting the values, we get

5.5 = 7 - 3(sd)

adding 3(sd) to each side, we get

5.5 + 3(sd) = 7-3(sd) + 3(sd)

or 5.5 + 3(sd) = 7

subtracting 5.5 on each side, we get

3(sd) = 7 -5.5 = 1.5

dividing both sides by 3, we get

sd = 1.5/3 = 0.5

So, standard deviation is 0.5

Now, we need to know that 6cm and 7.5 are located at how many standard deviation from the mean

we can write 6 as 7 - 2*(0.5) or mean - 2(sd)

similarly, 7.5 can written as 7 + 0.5 or mean + 1(sd)

So, it is clear that 6.5 is two standard deviation below mean and 7.5 is one standard deviation above mean

Using the empirical rule, 2 standard deviation below is 47.5% and 1 standard deviation above is 34%

So, total percent of value within 6. and 7.5 is 47.5+34 = 81.5%