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A researcher is wondering whether the drinking habits of adults in a certain region of the country are inthe same proportion as the general population of adults. Suppose a recent study stated that theproportion of adults who reported drinking once a week or less in the last month was 0.26. The nullhypothesis for this test is H0: p = 0.26 and the alternative hypothesis is Ha: p < 0.26. The researchercollected data from 150 surveys he handed out at a busy park located in the region. Check that theconditions hold so that the sampling distribution of the z-statistic will approximately follow the standardnormal distribution. Are the conditions satisfied? If not, choose the condition that is not satisfied.A) Yes, all the conditions are satisfied.B) No, the researcher did not collect a large enough sample.C) No, the population of interest is not large enough to assume independence.D) No, the researcher did not collect a random sample

Question

A researcher is wondering whether the drinking habits of adults in a certain region of the country are inthe same proportion as the general population of adults. Suppose a recent study stated that theproportion of adults who reported drinking once a week or less in the last month was 0.26. The nullhypothesis for this test is H0: p = 0.26 and the alternative hypothesis is Ha: p < 0.26. The researchercollected data from 150 surveys he handed out at a busy park located in the region. Check that theconditions hold so that the sampling distribution of the z-statistic will approximately follow the standardnormal distribution. Are the conditions satisfied? If not, choose the condition that is not satisfied.A) Yes, all the conditions are satisfied.B) No, the researcher did not collect a large enough sample.C) No, the population of interest is not large enough to assume independence.D) No, the researcher did not collect a random sample

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Solution

Para verificar si las condiciones se cumplen para que la distribución muestral del estadístico z siga aproximadamente la distribución normal estándar, debemos revisar las siguientes condiciones:

  1. Tamaño de la muestra (n): La muestra debe ser lo suficientemente grande. Una regla común es que tanto np np como n(1−p) n(1-p) deben ser mayores o iguales a 10.

    • En este caso, n=150 n = 150 y p=0.26 p = 0.26 .
    • np=150×0.26=39 np = 150 \times 0.26 = 39 .
    • n(1−p)=150×(1−0.26)=111 n(1-p) = 150 \times (1 - 0.26) = 111 .
    • Ambos valores son mayores que 10, por lo que esta condición se cumple.

  2. Independencia: Las observaciones deben ser independientes. Esto generalmente se asegura si la muestra es aleatoria y si la población es al menos 10 veces mayor que el tamaño de la muestra.

    • No se menciona el tamaño de la población, pero asumiremos que es suficientemente grande.
    • Sin embargo, la muestra no fue aleatoria, ya que se recolectó en un parque ocupado, lo que podría introducir sesgo.

  3. Distribución aproximada normal: Para que la distribución muestral del estadístico z siga una distribución normal estándar, las condiciones anteriores deben cumplirse.

Dado que la muestra no fue recolectada de manera aleatoria, la condición de independencia no se cumple. Por lo tanto, la respuesta correcta es:

D) No, el investigador no recolectó una muestra aleatoria.

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