Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

A factory produces baseballs. The weights of the baseballs produced are approximately normally distributed with a mean weight of 145 grams and a standard deviation of 2 grams. Official rules require the balls to weigh between 142 and 149 grams.

Recall that the proportion of items in an interval of an approximately normally distributed situation can be estimated by the area under the normal curve. A table can be used to determine the area under a normal curve bounded by an interval.

First, the relevant values need to be converted to a z-score. A value's z-score represents the number of standard deviations it is above the mean. In the baseball example, the value 147 grams has a z-score of 1 because it is 1 standard deviation above the mean. The value 140 grams has a z‑score of -2.5 because it is 2.5 standard deviations below the mean.

In general, a z-score can be found using

1. Find the z-score for 142 grams.
2. Find the z-score for 149 grams.
3. What value has a z-score of 1.45?
4. The table gives the area under the normal curve that is less than the given value. Shade the region that is given by the table for the area related to 142 grams.

   

   Normal distribution graph on a coordinate plane with no grid. Horizontal axis scale 140 to 150 by 2’s, labeled “baseball weight (grams)”. Vertical axis scale 0  point 0 2 to 0 point 1 4 by 0 point 0 2’s. The graph is a normal distribution curve with a mean of 145 and a standard deviation of 2.

5. Use the z-score for 142 grams and the table to find the area under the normal curve that is less than 142 grams.

6. Shade the region that is given by the table for the area related to 149 grams.

   

   Normal distribution graph on a coordinate plane with no grid. Horizontal axis scale 140 to 150 by 2’s, labeled “baseball weight (grams)”. Vertical axis scale 0  point 0 2 to 0 point 1 4 by 0 point 0 2’s. The graph is a normal distribution curve with a mean of 145 and a standard deviation of 2.

7. Use the z-score for 149 grams and the table to find the area under the normal curve that is less than 149 grams.
8. Use the two areas to find the area under the normal curve between 142 and 149 grams. Explain or show your reasoning.
9. What proportion of the baseballs that the factory makes is estimated to be within the official rules?