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SIZIF [17.4K]
3 years ago
15

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 f

ollow a normal distribution. Her average score is 225, with a standard deviation of 13. If during a typical week Susan bowls 16 games, what is the probability that her average score is more than 230? (A) 0.0620(B) 0.3520(C) 0.6480(D) 0.9382
Mathematics
2 answers:
notsponge [240]3 years ago
8 0

Answer:

There is a 6.18% probability that her average score is more than 230.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean \mu and standard deviation \sigma, a large sample size can be approximated to a normal distribution with mean \mu and standard deviation \frac{\sigma}{\sqrt{n}}

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

Her average score is 225, with a standard deviation of 13. This means that \mu = 225, \sigma = 13.

If during a typical week Susan bowls 16 games, what is the probability that her average score is more than 230?

This is 1 subtracted by the pvalue of Z when X = 230.

By the Central Limit Theorem, we have s = \frac{\sigma}{\sqrt{n}} = \frac{13}{4} = 3.25

Z = \frac{X - \mu}{s}

Z = \frac{230 - 225}{3.25}

Z = 1.54

Z = 1.54 has a pvalue of 0.9382. This means that there is a 1-0.9382 = 0.0618 = 6.18% probability that her average score is more than 230.

dimulka [17.4K]3 years ago
6 0

It would be nice if you spaced the question out properly

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