For Tommy Bench, a minor league pitcher, the times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of 2.5 seconds. If Bench performs 50 pitches, how many of them (to the nearest whole number) do you expect to have times-between-pitches to be longer than 40 seconds?
http://davidmlane.com/hyperstat/z_table.html
To find the number of pitches with times-between-pitches longer than 40 seconds, we need to calculate the probability and then multiply it by the total number of pitches.
First, we need to standardize the value of 40 seconds using the formula for z-score:
z = (x - μ) / σ
Where:
x = 40 seconds (value we want to standardize)
μ = mean of the distribution = 36 seconds
σ = standard deviation of the distribution = 2.5 seconds
Using the formula:
z = (40 - 36) / 2.5 = 4 / 2.5 = 1.6
Next, we need to find the probability of having a z-score greater than 1.6 using a standard normal distribution table or a calculator. The probability corresponds to the area under the curve to the right of the z-score.
Checking the standard normal distribution table, the area to the right of 1.6 is approximately 0.0548.
So, the probability of a pitch having a time-between-pitches longer than 40 seconds is 0.0548.
Finally, we can calculate the expected number of pitches that satisfy this condition:
Expected number = Probability * Total number of pitches
Expected number = 0.0548 * 50 = 2.74
To the nearest whole number, we expect around 3 pitches to have times-between-pitches longer than 40 seconds for Tommy Bench.