A grocery store owner rates the quality of her oranges on a

A grocery store owner rates the quality of her oranges on a 0 to 20 scale. She has 15 oranges, with ratings listed below. For this problem we will think of the 15 oranges as an entire population. The population mean rating is mu = 10 and the population standard deviation for the ratings is Rho= 4. What does the standard deviation tell us about variability with respect to the population mean ? Suppose you buy the orange with a rating of 14. Is this above or below average in quality? How far does this value deviate from the mean? How many standard deviations is this? The number of standard deviations that a value is from the mean Is called its Z-score. What is the Z- score of the orange with a rating of x = 14?

Solution

The Z-score of the orange with a rating of x=14, Z=(x-)/=(14-10)/4=4/4=1, here area under the curve is 68.26%.   The Z-score of the orange with a rating of x=18, Z=(x-)/=(18-10)/4=8/4=2, here the area under the curve is 95.44%.   The Z-score of the orange with a rating of x=4, Z=(x-)/=(4-10)/4=-6/4=-1.5, here Z value is negative, but by the definition of symmetry it becomes the area under the curve is 87%.The Z-score of the orange with a rating of x=10, Z=(x-)/=(10-10)/4=0/4=0, here the area under curve is 50% because the value of x==10.