Math 207 – Normal Distribution Examples

Example 1 (Using Table 3—that is, working directly with the z-values)

Consider the standard normal distribution, N(0, 1).

1. What is the area under the curve to the left of 1.35?

2. What is the area under the curve to the right of 1.35?

3. What is the area under the curve between –1.21 and 0.80?

Example 2

Suppose the distribution of scores on an economics exam is approximately normal with mean 68.2 points and standard deviation 12.5 points. Let the random variable, X, be the score of a randomly chosen exam. Then X has the N(68.2, 12.5) distribution.

1. Determine the probability that X is between 60 and 70 points.

2. Only students with scores in the top 10% of the distribution receive an A. What is the minimum score a student must get in order to receive an A?

Example 3

Agricultural scientists are working on developing an improved variety of Roma tomatoes. Marketing research indicates that customers are likely to bypass Romas that weigh less than 70 grams. The current variety of Roma plants produces fruit that average 74 grams, but 11% of the tomatoes are too small (i.e., weigh less than 70 grams). The weights of these tomatoes follow an approximate normal distribution.

1. What is the standard deviation of the weights of Romas now being grown?

2. Scientists hope to reduce the frequency of undersized tomatoes to no more than 4%. One way to accomplish this is to raise the average size of the fruit. If the standard deviation remains the same, what target mean should they have as a goal?