MATHEMATICS COLLEGE

At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed was 100 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. Assume that the statistician also gave us the information that the distribution of serve speeds was mound- shaped and symmetric. What percentage of the player's serves were between 115 mph and 145 mph

Answers

Answer 1
Answer:

Answer:

15.74% of the player's serves were between 115 mph and 145 mph

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

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

Z = (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 question, we have that:

\mu = 100, \sigma = 15

What percentage of the player's serves were between 115 mph and 145 mph

This is the pvalue of Z when X = 145 subtracted by the pvalue of Z when X = 115.

X = 145

Z = (X - \mu)/(\sigma)

Z = (145 - 100)/(15)

Z = 3

Z = 3 has a pvalue of 0.9987

X = 115

Z = (X - \mu)/(\sigma)

Z = (115 - 100)/(15)

Z = 1

Z = 1 has a pvalue of 0.8413

0.9987 - 0.8413 = 0.1574

15.74% of the player's serves were between 115 mph and 145 mph
Answer 2
Answer:

Final answer:

A total of 27% of the player's serves at the U.S. Open Tennis Championship were between 115mph and 145mph. This was found using the Empirical Rule which applies to a normal distribution of serve speeds.

Explanation:

This problem is a classic example of the use of the Empirical Rule in statistics. The Empirical Rule, also known as the 68-95-99.7 rule, applies to a normal distribution, which is a bell-shaped curve (mound-shaped and symmetric) as mentioned in the problem. This rule states that approximately 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Given that the mean serve speed is 100 mph and the standard deviation is 15 mph, serves of 115 mph are one standard deviation above the mean and serves of 145 mph are three standard deviations above the mean. Therefore, we are looking for the percentage of serves between these two values. According to the Empirical Rule, this would be 95% (coverage for up to 2 standard deviations) minus 68% (coverage for up to 1 standard deviation), which equals 27%. So, 27% of the player's serves were between 115 mph and 145 mph.

Learn more about Empirical Rule here:

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Answers

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Answer:

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Step-by-step explanation:

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__

Additional comment

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Answers

Answer:

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Step-by-step explanation:

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A city currently has 31,000 residents and is adding new residents steadily at the rate of 1200 per year. If the proportion of residents that remain after t years is given by S(t) = 1/(t + 1), what is the population of the city 7 years from now?

Answers

Answer:

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Given:

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Step-by-step explanation:

Let the initial population be  P_(i) = 31000

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Answers

Answer:

On a full tank of gas, Tommy can drive a distance:

D = 22 x 16 = 352 miles

Hope this helps!

:)

Answer:352 miles

Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:
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