Susan has been on a bowling team for 14 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 225, with a standard deviation of 13. What is the probability that in a one-game playoff, her score is more than 227?Multiple Choice0.55960.26760.73240.4244
Question
Susan has been on a bowling team for 14 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 225, with a standard deviation of 13. What is the probability that in a one-game playoff, her score is more than 227?Multiple Choice0.55960.26760.73240.4244
Solution
To solve this problem, we need to use the concept of Z-score in statistics. The Z-score is a measure of how many standard deviations an element is from the mean.
Step 1: Calculate the Z-score. The formula for the Z-score is Z = (X - μ) / σ, where X is the value of the element, μ is the mean, and σ is the standard deviation.
In this case, X = 227 (Susan's score), μ = 225 (the average score), and σ = 13 (the standard deviation).
So, Z = (227 - 225) / 13 = 0.154
Step 2: Look up this Z-score in the Z-table to find the probability. The Z-table tells us what percentage of the data lies below (to the left of) our Z-score.
For Z = 0.154, the Z-table gives us a value of 0.5596. This means that 55.96% of the data lies below a score of 227.
Step 3: Since we want to find the probability that her score is more than 227, we need to subtract this percentage from 1 (because the total probability is always 1).
So, 1 - 0.5596 = 0.4404
Therefore, the probability that Susan scores more than 227 in a one-game playoff is 0.4404 or 44.04%.
So, the closest answer among the options given is 0.4244.
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