The distribution of the number of viewers for the American Idol television show follows a normal distribution with a mean of 27 million and a standard deviation of 10 million.
A: Have between 29 and 37 million viewers?
B: Have at least 21 million viewers?
C: Exceed 43 million viewers?
You can play around with Z table stuff here:
http://davidmlane.com/hyperstat/z_table.html
To determine the probabilities for different viewer ranges in a normal distribution, we can use the standardization formula and the Z-table.
The standardization formula is: Z = (X - μ) / σ
Where:
- Z is the standardized score
- X is the value we want to analyze
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
First, let's calculate the standardized scores for each scenario:
A: Have between 29 and 37 million viewers?
Z1 = (29 - 27) / 10 ≈ 0.2
Z2 = (37 - 27) / 10 ≈ 1
B: Have at least 21 million viewers?
Z = (21 - 27) / 10 ≈ -0.6
C: Exceed 43 million viewers?
Z = (43 - 27) / 10 ≈ 1.6
Once we have the standardized scores, we can use the Z-table to find the probabilities.
A: To find the probability of having between 29 and 37 million viewers, we need to calculate the area under the normal distribution curve between Z1 and Z2. This corresponds to the probability of the event occurring.
B: To find the probability of having at least 21 million viewers, we need to calculate the area under the curve to the left of Z. This corresponds to the cumulative probability up to that point.
C: To find the probability of exceeding 43 million viewers, we need to calculate the area under the curve to the right of Z. This corresponds to 1 minus the cumulative probability up to that point.
To find these probabilities, you can refer to a Z-table, which provides the cumulative probability for different standardized scores. By looking up the values for each scenario, you can determine the respective probabilities.
Remember to convert any z-scores to their corresponding probabilities by referring to the standard normal distribution table.