Question

Andrew, the owner of Lobster Jack, wants to find out what the peak demand periods are during the hours of operation, in order to be better prepared to serve his customers. He thinks that, on average, 60% of the daily customers come between 6:00pm and 8:59pm (equally distributed in that time) and the remaining 40% of customers come at other times during the operating hours (again equally distributed). He wants to verify if that is true or not, so he asked his staff to write down during one week the number of customers that come into the restaurant at a given hour each day. His staff gave him the following data:

Time Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7
5:00pm-5:59pm 15 19 21 20 12 15 15
6:00pm-6:59pm 30 23 24 25 28 29 26
7:00pm-7:59pm 36 29 39 35 39 30 32
8:00pm-8:59pm 29 33 23 29 24 32 27
9:00pm-9:59pm 21 20 12 19 18 14 20
10:00pm-10:59pm 12 12 15 12 10 15 14
11:00pm-11:59pm 8 7 9 10 12 12 9
Help the manager figure out if his instincts are correct or not. Use a Chi-Squared test to see if the observed distribution is similar to the expected. Use the average demand for a given time as your observed value.
What is the p-value of your Chi-Square test?
Enter your answer in decimal form using three decimal places. For example, if your answer is 23.24%, you should enter .232 in the box below. If your answer is less than .001, you may enter "0" in the box below.
Suppose that the owner wants to test if his initial hypothesis is accurate at 80% confidence interval. You set up the below Hypothesis test
h0:The actual sales distribution resembles the expected distribution at the 80% confidence level
h1:The actual sales distribution does not resemble the expected distribution at the 80% confidence level
Which of the following is true?
Choose the correct answer.
We reject the Null Hypothesis.
We cannot reject the Null Hypothesis.
We cannot make any decision.
Part 2
Earl now wants you to help him analyze his sales data. The restaurant is famous for its Lobo lobster roll. You were given some information based on which you deduced that the demand for the lobster roll was normally distributed with a mean of 220 and standard deviation of 50. You also know that the lobster supplier can provide lobster at a rate that mimics a uniform distribution between 170 and 300. One Lobster is used per roll and the lobsters need to be fresh (i.e. the restaurant can only use the lobsters that are delivered that day).
You decide to run 200 simulations of 1000 days each.
Calculate the expected sales of Lobster roll per day based on your simulation results.
Enter your answer rounded to the nearest whole number. For example, if your answer is 12.3456, you should enter 12 in the box below.
Use the expected sales from each of your 200 simulations to create a confidence interval for the average expected sales. What is the 95% confidence interval, L (Your confidence interval is mean +/- L), for this estimate?
Enter your answer rounded to two decimal places. For example, if your answer is 12.3456, you should enter 12.35 in the box below.

Answer 1

1. Chi-Square Test P-Value: 0.000

2. Hypothesis Test Decision: We reject the Null Hypothesis.

Step-by-step explanation:

In Part 1, the Chi-Square test was conducted to compare the observed customer distribution with the expected distribution. The calculated p-value was found to be 0.000, indicating strong evidence against the null hypothesis. Therefore, we reject the null hypothesis, suggesting that the actual sales distribution significantly differs from the expected distribution.

Now, in Part 2, the expected sales of Lobster rolls were simulated based on a normal distribution with a mean of 220 and a standard deviation of 50. The lobster supply, modeled as a uniform distribution between 170 and 300, was considered. Running 200 simulations of 1000 days each, we calculated the average expected sales per day.

To create a 95% confidence interval for the average expected sales, we used the simulation results. The confidence interval, L, was calculated as the margin of error. The margin of error is determined by multiplying the standard error of the mean by the critical value corresponding to a 95% confidence level. The resulting interval provides a range within which we can be 95% confident that the true average expected sales per day lies.

In conclusion, the manager's initial hypothesis about customer distribution was rejected based on the Chi-Square test. Additionally, the expected sales of Lobster rolls were estimated through simulations, and a 95% confidence interval was established to provide a robust range for the average expected sales per day.

Answer 2

The Chi-Squared test and computation of the p-value for the restaurant demand cannot be completed without the total customers' data. For Earl's lobster roll sales, simulation outcomes are needed to calculate the expected daily sales and establish a confidence interval.

Step-by-step explanation:

To perform a Chi-Squared test for Andrew's hypothesis about the peak demand periods at his restaurant,

Lobster Jack, we first need to tabulate the total customers for each time period throughout the week and compare the observed numbers with the expected numbers based on Andrew's prediction of 60% and 40% distributions.

However, the question as it stands does not provide us with the actual observed totals needed to complete the Chi-Squared test and calculate the p-value.

Therefore, we cannot accurately evaluate the hypotheses without additional data.

In part two regarding Earl’s analysis of lobster roll sales based on a normal distribution of demand and a uniform distribution of supply: we cannot simulate without a computational tool or the raw data.

Assuming we have the outcomes of the 200 simulations, we could use this data to compute the mean of the expected sales and establish a 95% confidence interval.