There is sufficient evidence to conclude that the proportion of U.S. households that own at least one television set and have two or more sets is less than 0.83.
The solution to the hypothesis test problem:
Given:
P = 223/297 = 0.7475 (sample proportion of households with two or more television sets)
p = 0.83 (population proportion of households with two or more television sets)
α = 0.10 (level of significance)
Hypotheses:
H₀: p ≥ 0.83 (null hypothesis: the proportion of households with two or more television sets is not less than 0.83)
H₁: p < 0.83 (alternative hypothesis: the proportion of households with two or more television sets is less than 0.83)
Test statistic:
Z = (P - p) / √(p(1 - p) / n) = (0.7475 - 0.83) / √(0.83(1 - 0.83) / 297) = -1.5272
p-value:
p = P(Z < -1.5272) = 0.0625
Decision:
Since the p-value (0.0625) is less than the level of significance (α = 0.10), we reject the null hypothesis (H₀).
Conclusion:
There is sufficient evidence to conclude that the proportion of U.S. households that own at least one television set and have two or more sets is less than 0.83.
Complete question:
According to Nielsen Media Research, of all the U.S. households that owned at least one television set, 83% had two or more sets. A local cable company canvassing the town to promote a new cable service found that of the 297 households visited, 223 had two or more television sets. At =α0.10, is there sufficient evidence to conclude that the proportion is less than the one in the report? Do not round intermediate steps.