Final answer:
To compare customer satisfaction levels, we calculate a 99% confidence interval for the average difference in ratings using the provided data for Company X and Company Y, and applying the appropriate statistical formula for two independent samples.
Step-by-step explanation:
To compare customer satisfaction levels between two cable companies, we construct a 99% confidence interval for the difference in average satisfaction levels. We use the sample mean (\(\bar{x}\)), standard deviation (s), and sample size (n) for Company X and Company Y to calculate this interval.
The confidence interval formula for the difference between two means is:
\(\bar{x}_1 - \bar{x}_2 \pm z * \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)
Where:
- \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means for Company X and Y respectively.
- \(s_1\) and \(s_2\) are the sample standard deviations for Company X and Y.
- \(n_1\) and \(n_2\) are the sample sizes for Company X and Y.
- z is the z-score corresponding to the desired confidence level, which can be found from a standard normal distribution table.
Since we are looking for a 99% confidence interval, the z-score is approximately 2.576. Plugging in the values given:
\(3.51 - 3.24 \pm 2.576 * \sqrt{\frac{0.51^2}{174} + \frac{0.52^2}{355}}\)
Calculating this, the confidence interval provides the estimated range of the difference in average customer satisfaction levels between the two companies.