Mathematics, 27.07.2021 01:10 tori6886

A motel has a policy of booking as many as 150 guests in a building that holds 140. Past studies indicate that only 85% of booked guests show up for their room. Find the probability that if the motel books 150 guests, not enough seats will be available.

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Answer from: hemolelekeakua

0.0015 = 0.15% probability that if the motel books 150 guests, not enough seats will be available.

Step-by-step explanation:

Binomial probability distribution

Probability of exactly x successes on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

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

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ , if $np \geq 10$ and $n(1-p) \geq 10$ .

150 guests booked:

This means that n = 150

85% of booked guests show up for their room.

This means that p = 0.85

Is the normal approximation suitable:

np = 150(0.85) = 127.5

n(1-p) = 150(0.15) = 22.5

Both greater than 10, so yes.

Mean and standard deviation:

$\mu = E(X) = np = 150*0.85 = 127.5$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{150*0.85*0.15} = 4.3732$

Find the probability that if the motel books 150 guests, not enough seats will be available.

More than 140 show up, which, using continuity correction, is P(X 140 + 0.5) = P(X 140.5) , which is 1 subtracted by the p-value of Z when X = 140.5. So

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

$Z = \frac{140.5 - 127.5}{4.3732}$

Z = 2.97

Z = 2.97 has a p-value of 0.9985.

1 - 0.9985 = 0.0015.

0.0015 = 0.15% probability that if the motel books 150 guests, not enough seats will be available.

Answer from: Quest

42.4

step-by-step explanation:

convert to decimal

Answer from: Quest

c. x=2.4cm

step-by-step explanation:

similar triangles are triangles which have the same shape but not the same size. this means their angle measures are equal but their side lengths are not instead they are proportional. to solve, we will set up a proportion and solve.

a proportion is two equal ratios set equal to each other. we form the ratios by finding corresponding sides (sides which match to each other on both triangles by their position). out ratios will be one side of the small triangle over the corresponding side on the big triangle as shown below:

$\frac{little}{big} =\frac{side mn}{side ab}=\frac{side on}{side cb}$ .

$\frac{0.96}{1.2}=\frac{x}{3}$

to solve the proportion, we'll cross multiply by multiplying numerator with denominator across the equal sign.

$0.96(3)=1.2(x)\\2.88=1.2x\\\frac{2.88}{1.2}=\frac{1.2x}{1.2} \\2.4=x$

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