Question

The Museum of Science in Boston has an exhibit in which metal balls drop down a chute, bounce around, and wind up in one of 21 bins. After 1000 balls have dropped, the heights within the bins clearly follow a bell-shaped curve. The standard deviation is 3 bins.

About how many balls are in bins 1 through 17?

Answer 1

Refer to the diagram shown below.

Assume that approximately all the data is contained under the probability distribution curve within 3 standard deviations from the mean. It is actually 99.7%.
Therefore,
bin #1 corresponds to μ-3σ,
bin #11 corresponds to μ,
bin #21 corresponds to μ + 3σ.

Let x be the random variable for bin 17.
Then by interpolation,

`(x-\mu)/((\mu + 3\sigma )-\mu) = (17-11)/(21-11) \\\\ x-\mu = 1.8\sigma`

The z-score for x is

`z= (x-\mu)/(\sigma) = (1.8\sigma)/(\sigma) =1.8`

The probability corresponding to bin #17 is (from standard tables)
P(z≤1.8) = 0.964
Therefore if there are 1000 balls, the sum of the balls in bins 1 to 17 is
0.964 * 1000 = 96.4 ≠ 96 balls.

Answer: 96 balls