Question

Zainab runs her own business where she leads guided day-long tours for small groups of tourists in her city. Each tour has a maximum of 333 guests. Let XXX represent the number of customers Zainab gets on a randomly chosen day. Based on previous data, here is the probability distribution of XXX along with summary statistics: X=\text{\# of customers}X=# of customersX, equals, start text, \#, space, o, f, space, c, u, s, t, o, m, e, r, s, end text 000 111 222 333 P(X)P(X)P, left parenthesis, X, right parenthesis 0.200.200, point, 20 0.100.100, point, 10 0.400.400, point, 40 0.300.300, point, 30 Mean: \mu_X=1.8μ X ​ =1.8mu, start subscript, X, end subscript, equals, 1, point, 8 Standard deviation: \sigma_X\approx1.08σ X ​ ≈1.08sigma, start subscript, X, end subscript, approximately equals, 1, point, 08 It costs Zainab \$5$5dollar sign, 5 each day to pay for her own transportation to her business, and then each customer she gets pays her \$60$60dollar sign, 60. Let YYY represent Zainab's net gain on a randomly chosen day. What are the mean and standard deviation of YYY?

Answer 1

The ansa
μ 103 dollars

σ 64.8 dollars

Answer 2

An

μ Y=103 dollars

σ Y≈64.8 dollars

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Explanation:

We can find her income by multiplying the number of customers by 606060, and then subtracting her fixed transportation cost to get her net gain:

Y=60X-5Y=60X−5Y, equals, 60, X, minus, 5

How will this transformation impact the mean and standard deviation?

Hint #22 / 4

Effect on mean

Multiplying by 606060 and subtracting 555 both impact the mean:

\begin{aligned} \mu_Y&=60\left(\mu_X\right)-5 \\\\ &=60\left(1.8\right)-5 \\\\ &=108-5 \\\\ &=103 \end{aligned}

μ

Y

​

​

=60(μ

X

​

)−5

=60(1.8)−5

=108−5

=103

​

Hint #33 / 4

Effect on standard deviation

Multiplying by 606060 impacts the standard deviation, but subtracting 555 does not — adding or subtracting a constant only shifts data, which doesn't affect the spread.

\begin{aligned} \sigma_Y&=60\left(\sigma_X\right) \\\\ &\approx60\left(1.08\right) \\\\ &\approx64.8 \end{aligned}

σ

Y

​

​

=60(σ

X

​

)

≈60(1.08)

≈64.8

​