Question

PLEASE HELP!! 20 POINTS EACH!!
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Professional tennis player Novak Djokovic hits the ball extremely hard. His first serve speeds can be modeled by a normal distribution...
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Please help me with question 62

Answer 1

Answer:

See attachment

Explanation:

See attachment

Answer 2

The speeds of Novak Djokovic first serve are modeled by normal distribution.

• 5.48% of his first serves are greater than 120 mph
• 0.82% of his first serves are less than 100 mph
• 33.64% of his first serves are between 100 mph and 110 mph

The given parameters are:

$\mathbf{\mu = 112}$ --- mean

$\mathbf{\sigma = 5}$ ---- standard deviation

(a) How often he serves faster than 120 mph

This means that:

$\mathbf{x = 120}$

Start by calculating the z-score

$\mathbf{z = \frac{x - \mu}{\sigma}}$

$\mathbf{z = \frac{120 - 112}{5}}$

$\mathbf{z = \frac{8}{5}}$

$\mathbf{z = 1.6}$

So, the proportion of times is calculated using:

$\mathbf{P(x > 120) = P(z > 1.6)}$

Using z-score table, we have:

$\mathbf{P(x > 120) = 0.054799}$

Express as percentage

$\mathbf{P(x > 120) = 5.4799\%}$

Approximate

$\mathbf{P(x > 120) = 5.48\%}$

(b) Proportion of times he serves lesser than 100 mph

This means that:

$\mathbf{x = 100}$

Start by calculating the z-score

$\mathbf{z = \frac{x - \mu}{\sigma}}$

$\mathbf{z = \frac{100 - 112}{5}}$

$\mathbf{z = \frac{-12}{5}}$

$\mathbf{z = -2.4}$

So, the proportion of times is calculated using:

$\mathbf{P(x < 100) = P(z <- 2.4)}$

Using z-score table, we have:

$\mathbf{P(x < 100) = 0.0081975}$

Express as percentage

$\mathbf{P(x < 100) = 0.81975\%}$

Approximate

$\mathbf{P(x < 100) = 0.82\%}$

(c) Proportion of times he serves between 100 mph and 110 mph

This means that:

$\mathbf{x_1 = 100\ and\ x_2 = 110}$

In (b)

$\mathbf{z = -2.4}$, when $\mathbf{x =100}$

When x = 110, we have:

$\mathbf{z = \frac{110 - 112}{5}}$

$\mathbf{z = \frac{-2}{5}}$

$\mathbf{z = -0.4}$

So, the proportion of times is calculated using:

$\mathbf{P(100<x < 110) = P(-2.4<z <- 0.4)}$

This gives:

$\mathbf{P(100<x < 110) = P(z <- 0.4) - P(z<-2.4)}$

Using z-score table, we have:

$\mathbf{P(100<x < 110) = 0.34458 - 0.0081975}$

$\mathbf{P(100<x < 110) = 0.3363825}$

Express as percentage

$\mathbf{P(100<x < 110) = 33.63825\%}$

Approximate

$\mathbf{P(100<x < 110) = 33.64\%}$

Hence, 33.64% of his first serves are between 100 mph and 110 mph

Read more about probabilities of normal distribution at:

https://brainly.com/question/6476990