Use the following information to answer questions 1-3. Alex tracked how many cups of coffee he drank each day for a month and then compiled the results into a probability distribution, as shown below Number of Cups of Coffee X 13 Relative Frequency, P(X) 0.0240423 0.2910.1720.090 Determine the median of alex’s probability distribution.
To find the median of a probability distribution, we need to find the value of x for which the cumulative probability is 0.50.
One way to find the cumulative probabilities is to start by adding up the relative frequencies from smallest to largest, keeping track of the running total.
Starting with the smallest value of x:
- For X = 0, P(X) = 0.0240423, so the cumulative probability up to this point is 0.0240423.
- For X = 1, P(X) = 0.2910, so the cumulative probability up to this point is 0.0240423 + 0.2910 = 0.3150423.
- For X = 2, P(X) = 0.172, so the cumulative probability up to this point is 0.3150423 + 0.172 = 0.4870423.
- For X = 3, P(X) = 0.090, so the cumulative probability up to this point is 0.4870423 + 0.090 = 0.5770423.
Since the cumulative probability at X = 2 is less than 0.50 and the cumulative probability at X = 3 is greater than 0.50, we know that the median must be between 2 and 3 cups of coffee per day.
To estimate the exact value, we can use linear interpolation. The difference between the cumulative probabilities at X = 2 and X = 3 is 0.5770423 - 0.4870423 = 0.09. We want to find how far along this difference we need to go to reach a cumulative probability of 0.50.
0.50 - 0.4870423 = 0.0129577
So we need to go about 0.0129577/0.09 = 0.143975 or approximately 14.4% of the way from X = 2 to X = 3.
To find the corresponding value of X, we add 0.143975 times the difference between X = 2 and X = 3 to X = 2:
2 + 0.143975(1) = 2.143975
Therefore, the median of Alex's probability distribution is approximately 2.14 cups of coffee per day.