On some pianos, the average distance from one white key to the next is 2.39 centimeters. How many of your classmates could reach two notes that are 9 keys apart (21.5 cm) on a keyboard using only one hand?

1. Stretch your fingers apart as wide as you can, and measure the farthest distance from your thumb to your smallest finger. Round your measurement to the nearest tenth of a centimeter.
2. Your teacher will collect the measurements from the class. Draw a dot plot or histogram from the class data.

   

3. Describe the distribution you drew using terms such as “symmetric,” “approximately symmetric,” “skewed left,” “skewed right,” “approximately uniform,” “uniform,” “bell-shaped,” or “bimodal.” Estimate the center of your distribution.
4. How would you use your distribution to determine how many people in the class can reach the two notes 9 keys apart?

Manufacturers of butter make sticks of butter that weigh 110 grams on average. A manufacturer suspects the machine that forms the sticks of butter may have a problem, so they weigh each stick of butter the machine produces in an hour.

The same data are summarized in this histogram.

Histogram. The vertical axis is labeled from 0 to 240 by 20’s. The horizontal axis, labeled weight, grams, has bin widths of .5, starting at 107. Beginning at 107 up to, but not including 107.5, height of bar at each interval is 5, 17, 52, 118, 172, 232, 219, 172, 95, 57, 23, 8, 1, 0, 0.

​​​​Although this information is useful, it might be more helpful to know the proportion of sticks of butter in each weight interval rather than the actual number of sticks in that weight interval.

1. Complete the table by dividing each frequency value by the total number of sticks of butter in the data set. Round each value to 4 decimal places.

2. A relative frequency histogram is a histogram in which the height of each bar is the relative frequency. Since the heights of the bars are found by dividing each height by the total number of sticks of butter, the shape of the distribution is the same as a regular histogram, but the labels on the -axis are changed. Label the -axis with the correct values for each mark.

   

   Histogram. The vertical axis labels are blank. The horizontal axis, labeled weight, grams, has bin widths of .5, starting at 106. The bar heights are shaped in a bell curve.

3. The manufacturer believes they should replace parts of the machine if more than 25% of the sticks of butter are more than 1 gram away from the intended value of 110 grams.
   1. Indicate on the relative frequency histogram the bars that correspond to sticks of butter that are more than 1 gram away from the intended weight of a stick of butter.
   2. Should this machine be replaced? Explain or show your reasoning.

These curves represent normal distributions with different means and standard deviations. What do you notice?

A bell-shaped distribution with the horizontal axis labeled from 0 to 20 by 2’s. The peak of the bell is centered at 10. The vertical axis labeled from 0 to 0.5 by 0.1’s. The bell curve is skinny, with most of the curve between 8 and 12.

A bell-shaped distribution with the horizontal axis labeled from 0 to 20 by 2’s. The peak of the bell is centered at 10. The vertical axis labeled from 0 to 0.5 by 0.1’s. The bell curve is skinny, with most of the curve between 8 and 12.

A bell-shaped distribution with the horizontal axis labeled from 0 to 20 by 2’s. The peak of the bell is centered at 12. The vertical axis labeled from 0 to 0.5 by 0.1’s. The bell curve is skinny, with most of the curve between 10 and 14.

A bell-shaped distribution with the horizontal axis labeled from 0 to 20 by 2’s. The peak of the bell is centered at 8. The vertical axis labeled from 0 to 0.5 by 0.1’s.

A bell-shaped distribution with the horizontal axis labeled from 0 to 20 by 2’s. The peak of the bell is centered at 10. The vertical axis labeled from 0 to 0.5 by 0.1’s.