Question

It is claimed that the average score earned by high school graduates in the state of Virginia on the SAT test exceeds 500 points. A sample of 70 high school graduates in the state of Virginia obtained an average of 530 points. We assume the population standard deviation to be 125 points. Test at the 5% level of significance the statement made about the average points on the SAT test described in the previous case.

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Answer to a math question It is claimed that the average score earned by high school graduates in the state of Virginia on the SAT test exceeds 500 points. A sample of 70 high school graduates in the state of Virginia obtained an average of 530 points. We assume the population standard deviation to be 125 points. Test at the 5% level of significance the statement made about the average points on the SAT test described in the previous case.

Expert avatar
Andrea
4.5
44 Answers
Para probar si el promedio de puntos obtenidos por graduados de bachillerato en el estado de Virginia en la prueba SAT excede los 500 puntos, utilizamos la prueba de hipótesis.

Dados:

- $\mu$: promedio de puntos obtenidos por graduados de bachillerato en el estado de Virginia para la prueba SAT.
- $n = 70$: tamaño de la muestra.
- $\bar{x} = 530$: promedio de puntos obtenidos en la muestra.
- $\sigma = 125$: desviación estándar poblacional.
- Nivel de significancia $\alpha = 0.05$.

Las hipótesis nula y alternativa serán:

$H_0: \mu \leq 500$

$H_1: \mu > 500$

Para calcular el estadístico de prueba, utilizamos la fórmula:

Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}

Donde:

- $\bar{x}$ es el promedio de la muestra,
- $\mu_0 = 500$ es el promedio afirmado en la hipótesis nula,
- $\sigma$ es la desviación estándar poblacional, y
- $n$ es el tamaño de la muestra.

Sustituyendo los valores dados:

Z = \frac{530 - 500}{\frac{125}{\sqrt{70}}} \approx \frac{30}{14.978} \approx 2.003

Para un nivel de significancia del 5% y considerando que nuestra hipótesis alternativa es del tipo $>$ (una cola), buscamos el valor crítico de la distribución normal estándar (z-score) que deja un área de cola derecha de 0.05, que es aproximadamente 1.645.

Como 2.003 > 1.645, rechazamos la hipótesis nula. Por lo tanto, hay suficiente evidencia para afirmar que el promedio de puntos obtenidos por graduados de bachillerato en el estado de Virginia para la prueba SAT excede los 500 puntos.

$\boxed{\text{Respuesta: Con un nivel de significancia del 5%, se rechaza la hipótesis nula. Hay evidencia suficiente para afirmar que el promedio de puntos obtenidos excede los 500 puntos.}}$

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