Una empresa que fabrica juguetes electrónicos afirma que las...

Una empresa que fabrica juguetes electrónicos afirma que las baterías que utiliza en sus productos duran un promedio de 30 horas. Para mantener este promedio se prueban 16 baterías cada mes. Si el valor $t$ calculado cae entre $-t_{0.025}$ y $t_{0.025}$, la empresa queda satisfecha con su afirmación. ¿Qué conclusiones debería sacar la empresa a partir de una muestra que tiene una media de $\bar{x}=27.5$ horas y una desviación estándar de $s=5$ horas? Suponga que la distribución de las duraciones de las baterías es aproximadamente normal.

Key Concepts

Hypothesis Testing

This concept involves formulating a null hypothesis (often a statement of no effect or default condition, such as the battery life being 30 hours) and an alternative hypothesis (suggesting a deviation from the null). In this case, the testing procedure is used to decide whether sample data provides enough evidence to reject the company’s claim regarding battery life.

Student's t-Test

The t-test is used when the population standard deviation is unknown and the sample size is small. It helps assess whether the observed sample mean differs significantly from the claimed population mean by taking into account the variability within the sample.

t Distribution

The t distribution is similar to the normal distribution but has heavier tails, which provides a more accurate reflection of variability when dealing with small samples. It is crucial for determining the critical values against which the computed t statistic is compared.

Critical Values and Significance Level

Critical values from the t distribution define the boundaries of the acceptance region for the null hypothesis at a given significance level. If the computed test statistic falls outside these boundaries, it indicates sufficient evidence to reject the null hypothesis.

Degrees of Freedom

Degrees of freedom, calculated as the sample size minus one for a one-sample t-test, are used in conjunction with the t distribution to obtain the correct critical values. This adjustment reflects the amount of independent information in the sample for estimating the standard error.

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