Final answer:
Hypothesis testing is used to determine if the proportion of U.S. households that own a dog differs from 30%. Hypotheses are formulated, a test statistic is calculated, with the evidence leading to a decision about the null hypothesis. The conclusion reflects the outcome of the test in relation to the original claim.
Step-by-step explanation:
Hypothesis Testing for Proportions
In the described scenario, a hypothesis test is performed to determine if the true proportion of U.S. households that own a dog has changed from the claimed 30%.
Part A: Hypotheses
The null hypothesis (H0) is that the proportion of households that own a dog is 30% (p = 0.3). The alternative hypothesis (Ha) is that the proportion of households that own dogs is not 30% (p ≠ 0.3).
Part B: Test Statistic
The test statistic is calculated using the formula for a proportion: z = (p' - p) / √[p(1-p)/n], where p' is the sample proportion, p is the hypothesized proportion, and n is the sample size. In this case, p' = 80/210 ≈ 0.3810, n = 210, and p = 0.3. The calculated z-value will indicate how many standard deviations the sample proportion is from the hypothesized proportion.
Part C: Critical Value/P-value
With an alpha level of 0.05, the critical z-values for a two-tailed test are approximately ±1.96. Alternatively, the p-value can be compared to alpha to determine if the null hypothesis should be rejected.
Part D: Decision
Based on the test statistic and the corresponding p-value or critical value, a decision is made whether to reject the null hypothesis or not.
Part E: Conclusion
The conclusion involves interpreting the statistical decision in the context of the problem. If the null hypothesis is rejected, we may conclude that there is enough evidence to believe that the proportion of U.S. households owning a dog is different from the claimed 30%.