The manufacturer of the ColorSmart-5000 television set claims 95 percent of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 420 consumers who have owned a ColorSmart-5000 television set for five years. Of these 420 consumers, 303 say their ColorSmart-5000 television sets did not need a repair, whereas 117 say their ColorSmart-5000 television sets did need at least one repair. (a) Find a 99 percent confidence interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair.

To calculate the 99% confidence interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair, we'll use the formula for the confidence interval for a proportion:

Confidence Interval = p̂ ± (z * √(p̂(1 - p̂) / n))

Where: - p̂ is the sample proportion - z is the z-score corresponding to the desired confidence level - n is the sample size

Step 1: Calculate the sample proportion (p̂). p̂ = (number of success) / (sample size) = 303 / 420

Step 2: Find the z-score corresponding to a 99% confidence level. A 99% confidence level corresponds to approximately 2.576 standard deviations from the mean on both sides if the distribution of the sample proportion is normal.

Step 3: Calculate the standard error of the proportion using: Standard Error (SE) = √(p̂(1 - p̂) / n)

Step 4: Calculate the margin of error using: Margin of Error (ME) = z * SE

Step 5: Calculate the confidence interval using: Confidence Interval = p̂ ± ME

Now we do the math:

p̂ = 303 / 420 ≈ 0.7214

z (for 99%) ≈ 2.576

SE = √(0.7214 * (1 - 0.7214) / 420) ≈ √(0.7214 * 0.2786 / 420) ≈ √(0.20098884 / 420) ≈ √0.000478545 ≈ 0.02187

ME = 2.576 * 0.02187 ≈ 0.05633

Now add and subtract the margin of error from the sample proportion:

Lower limit = p̂ - ME ≈ 0.7214 - 0.05633 ≈ 0.66507 Upper limit = p̂ + ME ≈ 0.7214 + 0.05633 ≈ 0.77773

Therefore, the 99% confidence interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair is (0.66507, 0.77773) or approximately 66.5% to 77.8%.

Extra: In statistics, a confidence interval is a range of values used to estimate a population parameter. The confidence level indicates the degree of certainty we have that the interval contains the true population parameter. A 99% confidence interval implies that if we were to take many random samples and compute a confidence interval for each sample, then approximately 99% of the intervals will contain the population parameter we are estimating.

For a proportion, as in this example, the confidence interval provides a range within which we believe the true proportion lies, based on the sample data. It is important to note that the confidence interval is just an estimate based on the sample and does not guarantee the true parameter lies within that interval. The larger the sample and the closer the confidence level is to 100%, the wider the confidence interval will be, reflecting a greater level of uncertainty.

Moreover, when the sample proportion is being used to estimate the population proportion, the sampling distribution of the sample proportion can often be approximated by a normal distribution if the sample size is large enough. This allows us to use the z-score in our calculations. The rule of thumb for this normal approximation to be valid is that both np̂ and n(1 - p̂) should be greater than 5, which is satisfied in the scenario provided.