Final answer:
To determine if there is a significant difference between the distribution of the number of televisions of far western U.S. families and the American population, a goodness-of-fit test using the chi-square statistic is used. If the chi-square value, which should be right-tailed, is high and the p-value is less than 0.01, the null hypothesis is rejected, indicating a significant difference at the 1 percent significance level.
Step-by-step explanation:
When assessing whether there is a statistically significant difference between observed frequencies and expected frequencies, a goodness-of-fit test is often used. This involves calculating a chi-square statistic, which helps determine if there is a significant difference between the two sets of frequencies. In this case, the test will help us determine if the distribution of the number of televisions in far western U.S. families differs from that of the American population.
At a 1 percent significance level (0.01), a high chi-square statistic implies that the observed data does not fit the expected distribution, leading to a rejection of the null hypothesis. The null hypothesis usually states that there is no significant difference between the observed and expected frequencies. The test statistic is always right-tailed, meaning that we are looking for values that lie in the upper end of the chi-square distribution.
If the p-value obtained from the chi-square test is less than 0.01, we would conclude that there is a statistically significant difference between the observed and expected distributions. It is also important to verify that each expected frequency is at least five to ensure the validity of the test.