Una librería recién inaugurada ofrece, además de la consulta de
libros, los servicios de cafetería.
Para una próxima exposición en la Feria del Libro, la empresa desea
fabricar camisetas promocionales.
La fábrica textil planea tres tallas: L,
XL y XXL.
Las alturas de los compradores siguen una distribución normal:
\[ X \sim N(\mu = 165.4, \sigma = 8.3) \]
\[ \begin{aligned} P(L) &= P(X \leq 161) \\ P(XL) &= P(161 < X \leq 179) \\ P(XXL) &= P(X > 179) \end{aligned} \]
Estandarizando:
\[ Z = \frac{X - \mu}{\sigma} \]
\[ Z_L = \frac{161 - 165.4}{8.3} = -0.53, \quad Z_{XL} = \frac{179 - 165.4}{8.3} = 1.64 \]
\[ P(Z \leq -0.53) = 0.2981, \quad P(Z \leq 1.64) = 0.9495 \]
\[ \begin{aligned} P(L) &= 0.2981 \\ P(XL) &= 0.9495 - 0.2981 = 0.6514 \\ P(XXL) &= 1 - 0.9495 = 0.0505 \end{aligned} \]
\[ \boxed{ P(L) = 29.8\%, \quad P(XL) = 65.1\%, \quad P(XXL) = 5.1\% } \]
\[ P(L) = 0.15, \quad P(XL) = 0.63, \quad P(XXL) = 0.22 \]
\[ P(X \leq x_1) = 0.15, \quad P(X \leq x_2) = 0.78 \]
\[ Z_1 = -1.04 \Rightarrow x_1 = 165.4 + (-1.04)(8.3) = 156.76 \] \[ Z_2 = 0.77 \Rightarrow x_2 = 165.4 + (0.77)(8.3) = 171.79 \]
\[ \boxed{ \text{Talla L: } X \leq 156.8, \quad \text{Talla XL: } 156.8 < X \leq 171.8, \quad \text{Talla XXL: } X > 171.8 } \]
\[ p = P(X \leq 163) = P\left(Z \leq \frac{163 - 165.4}{8.3}\right) = P(Z \leq -0.29) = 0.3859 \]
Sea \(Y \sim Binomial(n = 5, p = 0.3859)\):
\[ P(Y \geq 2) = 1 - [P(Y=0) + P(Y=1)] \]
\[ P(Y=0) = (0.6141)^5 = 0.0868, \quad P(Y=1) = 5(0.3859)(0.6141)^4 = 0.2725 \]
\[ P(Y \geq 2) = 1 - (0.0868 + 0.2725) = 0.6407 \]
\[ \boxed{P(Y \geq 2) = 0.6407 \Rightarrow 64.1\%} \]
\[ P(160 < X < 162) = P(X < 162) - P(X < 160) \]
\[ Z_1 = \frac{160 - 165.4}{8.3} = -0.65, \quad P(Z_1) = 0.2578 \] \[ Z_2 = \frac{162 - 165.4}{8.3} = -0.41, \quad P(Z_2) = 0.3409 \]
\[ P(160 < X < 162) = 0.3409 - 0.2578 = 0.0831 \]
\[ 30,000 \times 0.0831 = 2,493 \]
\[ \boxed{\text{Aproximadamente 2,493 camisetas entre 160 y 162 cm.}} \]