Final answer:
Option b, with a 99% margin of error of $1.88, is the closest to the calculated margin of error of $1.932 using a known standard deviation of $6 and a sample size of 64. Without the sample mean, we cannot provide the exact confidence interval, but it would have the form (x - 1.88, x + 1.88) for a given sample mean x.
Step-by-step explanation:
To determine which option (a, b, c, or d) correctly represents the 99% confidence interval for the average amount spent for lunch, we first need to establish the margin of error using the known population standard deviation (s = $6) and sample size (n = 64 customers).
The margin of error (EBM) for a confidence interval can be calculated using the formula EBM = Z * (σ/√n), where Z is the z-value corresponding to the desired confidence level (here 99%), σ is the known population standard deviation, and n is the sample size.
Looking up the z-value for 99% confidence, we typically get approximately 2.576. Using this z-value, the margin of error (EBM) would be:
EBM = Z * (σ/√n) = 2.576 * (6/√64) = 2.576 * (6/8) = 2.576 * 0.75 = $1.932.
The margin of error closest to the computed value is in option b (EBM = $1.88), which indicates this is the best option, assuming all else is equal and no other context such as the sample mean is provided.
The confidence interval is computed as (point estimate - margin of error, point estimate + margin of error). Since we don't have the sample mean provided, we cannot complete this interval. However, with a margin of error of $1.88, the interval would look like this when the sample mean (x) is known: (x - 1.88, x + 1.88).