Final answer:
To construct the 96% confidence interval for the mean bill of customers at a fast-food restaurant with a known population standard deviation, use the normal distribution and the z-score for the desired confidence level.
Step-by-step explanation:
The student is asking about constructing a confidence interval for the population mean of customer bills at a fast-food restaurant based on a sample. To build a 96% confidence interval when the population standard deviation is known, the normal distribution is used because, according to the Central Limit Theorem, the distribution of sample means will be approximately normal if the sample size is large enough (usually n > 30). Given the sample mean () is $13.47, the population standard deviation () is $2.54, and the sample size (n) is 63, we can calculate the confidence interval using the z-score corresponding to the desired confidence level.
To find the 96% confidence interval, we first determine the z-score that cuts off the upper 2% of the standard normal distribution (Z₀.9600), which is approximately 2.05. The margin of error (E) is then calculated as:
E = Z₀.9600 / / \sqrt{n}
E = 2.05 ($2.54) / \sqrt{63}
Next, we add and subtract the margin of error from the sample mean to get our confidence interval:
Lower bound = - E = $13.47 - E
Upper bound = + E = $13.47 + E
By completing these calculations, which are not present in this JSON object, the student will get the 96% confidence interval for the mean bill amount for all customers.