To win a prize at the county fair, the diameter of a tomato must be greater than 4 inches. Also, we are told that the diameters of of tomatoes grown in a special soil are normally distributed. We are asked to find the probability that a tomato grown in a special soil will be a winner, which is also the probability that its diameter will be greater than 4 inches.
P(Diameter>4)=?
We will first draw the probability distribution. The mean diameter is equal to 3.2 inches and the standard deviation is 0.4 inches. Let's find the diameters that are one, two, and three standard deviations away from the mean. For convenience, the mean will be represented by the letter m and the standard deviation will be represented by s.
m-3s
m-2s
m-s
m
m+s
m+2s
m+3s
Substitute
3.2-3( 0.4)
3.2-2( 0.4)
3.2- 0.4
3.2
3.2+ 0.4
3.2+2( 0.4)
3.2+3( 0.4)
Simplify
2
2.4
2.8
3.2
3.6
4
4.4
Now, let's draw vertical lines with the calculated values.
Finally, we can sketch the normal curve. The highest point of the curve should be at the mean, 3.2
The normal curve is divided into sections of standard deviation widths. Let's label the percentages of each section.
Now, let's highlight the parts of the graph representing tomatoes with diameters greater than 4 inches.
We can see that 2.35 % + 0.15 %=2.5 % of the tomatoes have diameters greater than 4 inches. This means that the probability that a tomato grown in the special soil will be a winner is also 2.5 %.
