Sanjay Thomas decided to run a simulation model of the earnings of the ABC Restaurant for the month of January with 400 trials. The observed sample mean of earnings for the month of January was $11,786 and the observed sample standard deviation of earnings was $8,236.

1. Construct a 99% condence interval for the mean monthly earnings of the restaurant.

2. How many trials does Sanjay need to run in order to predict the mean monthly earnings of the restaurant with an accuracy of plus or minus $200, with a condence level of 99%?

3. Sanjay is meeting with his statistician friend, Lucy Mathias. Lucy, based on her own analysis, declares: 'The 95% condence interval for the mean monthly earnings for January is [$11,000; $12,000]'. Are the following remarks of Sanjay TRUE or FALSE? (If you choose FALSE, please provide a brief (one sentence) explanation. No calculations are required.)

(a) 'This means that if I were to run a new simulation for monthly earnings in January, there is a 95% chance that the observed sample mean of my simulation would be between $11,000 and $12,000.'

(b) 'This means that if I were to run one simulation of 100 trials, then in only 5 trials, the value obtained for the monthly earnings would not be between $11,000 and $12,000'. Sanjay then asked Lucy how many trials she had used in her simulation, to which Lucy replied that she had used 900 trials. Are the following remarks of Sanjay TRUE or FALSE? (If you choose FALSE, please provide a brief (one sentence) explanation. No calculations are required.)

(c) 'I noticed that your condence interval has a range of $12,000 - $11,000= $1,000. So, if I were to run a simulation with 900 trials as well, I would also obtain a 95% condence interval with a range of $1,000'.

(d) 'Given the results of your simulation, a 99% condence interval for the mean monthly earnings of January would have a range larger than $1000'.

4. Sanjay's simulation model assumed that the number of meals served in a month obeys a Normal distribution with a mean of 3,000 meals and a standard deviation of 1,000 meals. When he did the calculations for the sample of 400 trials, he discovered that the observed sample mean of meals served in the month of January was 3,096 meals. Sanjay wondered if 3,096 was suciently dierent from 3,000 to suggest that there might be a software bug in the model. What is the likelihood that the observed sample mean differs from the true mean by more than 96 meals/month?