Write both the null and alternative hypothesis. Use the correct notation for full credit.
The proportion of households without a computer is 16%
The Correct Answer and Explanation is :
To test the proportion of households without a computer, we would be formulating hypotheses about this population parameter.
Null Hypothesis (H₀):
The null hypothesis represents the status quo or a statement of no effect. In this case, it states that the proportion of households without a computer is equal to 16%. Mathematically, it would be written as:
[ H_0: p = 0.16 ]
Where:
- ( p ) represents the population proportion of households without a computer.
Alternative Hypothesis (H₁):
The alternative hypothesis represents a statement that contradicts the null hypothesis. It is the claim we seek to test, either showing that the proportion is different from 16%, less than 16%, or greater than 16%. Since we don’t have any specific direction in the question, we will use a two-tailed test. Therefore, the alternative hypothesis will suggest that the proportion of households without a computer is not equal to 16%. This can be written as:
[ H_1: p \neq 0.16 ]
Where:
- ( p ) represents the population proportion of households without a computer.
Explanation:
In hypothesis testing, we are typically trying to determine if the sample data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis. In this case, the null hypothesis assumes that the proportion of households without a computer is exactly 16%, while the alternative hypothesis suggests that it could be any proportion different from 16%.
- The null hypothesis assumes there is no significant difference between the observed proportion and the expected proportion (16%). If the sample data provides sufficient evidence to support this, we will not reject the null hypothesis.
- The alternative hypothesis, on the other hand, claims that there is a significant difference from the 16% proportion. If the sample data shows a significant difference from this value, the null hypothesis may be rejected in favor of the alternative hypothesis.
We would typically use statistical methods such as a z-test for proportions to test the hypotheses, calculating a test statistic and comparing it to critical values or a p-value to make a decision about whether to reject the null hypothesis. The outcome of the hypothesis test would depend on the sample data, and the decision would guide further research or conclusions on the matter.