Final answer:
To calculate a 99 percent confidence interval for the proportion of ColorSmart-5000 television sets that lasted at least five years without needing a repair, one requires the sample proportion and sample size. Using the formula CI = π_hat ± Z * SE, where SE is the standard error and Z is the Z-score for the 99% confidence level, the confidence interval can be determined.
Step-by-step explanation:
To find a 99 percent confidence interval for the proportion of ColorSmart-5000 television sets that have lasted at least five years without needing a single repair, we would need the sample proportion (π_hat), the number of TV sets surveyed (n), and the standard error of the proportion.
The standard error can be calculated using the formula for the standard error of a proportion SE = sqrt[(π_hat * (1 - π_hat)) / n], where π_hat is the sample proportion and n is the sample size. After computing the standard error, the confidence interval can be found by taking the sample proportion and adding and subtracting the value of the Z-score corresponding to the desired confidence level (which can be found in a Z-table) multiplied by the standard error. The formula is CI = π_hat ± Z * SE.
Assuming we have the required sample data, which is not provided in the question, we would insert the values into the formula. If the sample proportion of TV sets that lasted at least five years without repair was found to be 0.75 (for example) out of a survey of 200 TV sets, we would calculate the standard error and then use the Z-score for a 99% confidence which is approximately 2.576, to find the confidence interval.
Without the actual sample data, we cannot compute the exact confidence interval. However, this explanation provides the method to calculate it once the sample data is available.