Mathematics, 06.04.2020 16:56 kordejah348

A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of $10.50 and a standard deviation of $5.20. He usually waits on about 50 parties over a weekend of work. a) Estimate the probability that he will earn at least $600. b) How much does he earn on the best 1% of such weekends?

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Answer from: Scienceissofun6453

(a) The probability that he will earn at least $600 is 0.0212.

(b) The amount of tip the waiter earns on the best 1% of such weekends is $610.67.

Step-by-step explanation:

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sum of values of X, i.e ∑X, will be approximately normally distributed.

Then, the mean of the distribution of the sum of values of X is given by,

$\mu_{X}=n\mu$

And the standard deviation of the distribution of the sum of values of X is given by,

$\sigma_{X}=\sqrt{n}\sigma$

The random variable X can be defined as the tips he receiver per order.

The average tip received by the waiter is, μ = $10.50.

The standard deviation of the tip received by the waiter is, σ = $5.20.

The waiter usually waits on about n = 50 parties over a weekend of work.

So, the distribution of the total tip earned by the waiter is:

$\sum X\sim N(\$525,\ \$36.77)$ .

(a)

Compute the probability that he will earn at least $600 as follows:

$P(\sum X\geq 600)=P(\sum X600-0.5)$

$=P(\sum X599.5)\\$

$=P(\frac{\sum X-\mu_{X}}{\sigma_{X}}\frac{599.5-525}{36.77})$

$=P(Z2.03)\\=1-(Z$

*Use a z-table for the probability.

Thus, the probability that he will earn at least $600 is 0.0212.

(b)

Let be a be the amount of tip the waiter earns on the best 1% of such weekends.

That is,

P (∑X > a) = 0.01

⇒ P (∑X < a) = 0.99

⇒ P (Z < z) = 0.99

The value of z for the above probability is:

z = 2.33.

Compute the value of a as follows:

$z=\frac{a-\mu_{X}}{\sigma_{X}}$

$2.33=\frac{a-525}{36.77}$

$a=525+(2.33\times 36.77}\\a=610.6741\\a\approx610.67$

Thus, the amount of tip the waiter earns on the best 1% of such weekends is $610.67.

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.

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

Answer from: Quest

Main idea on left, details on right

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A waiter believes the distribution of his tips has a model that is slightly skewed to the right, wit...