Question

The restaurant owner 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.

The owner 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. 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?

Answer 1

Solution

Back-up Theory

Goodness of Fit

Let Oi and Ei be respectively the observed and expected frequencies of the ith class, i = 1 to k, k being the number of classes given.

Hypotheses:

Null: H₀: Observed frequencies of are in accordance with the given expected frequencies. Vs

Alternative H_A: H₀ is false

Test Statistic:

χ2 = ∑[i = 1,k]{(Oi - Ei)²/Ei},

Distribution, Significance Level, α, Critical Value, p-value

Under H₀, χ² ~ χ²_{k – s}, Chi-square distribution with degrees of freedom = k – s,where k = number of classes and s =number of parameters estimated.

p-value = P(χ²_{k – s} > χ²_cal)

Given significance level = α , critical value = χ²_crit = upper α% of χ²_{k - s, α}

Critical value and p-value obtained using Excel Function: Statistical CHIINV and CHIDIST are as shown in the above table.

Decision

Since, χ2_cal > < χ2_crit, or equivalently, since p-value < > α, H₀ is rejected/accepted

Now to work out the solution,

Time Interval	6 to 8:59pm	Other	Total
Oi	622	398	1020
pi	0.6	0.4	1
Ei = 1020 x pi	612	408	1020
χ2	0.1634	0.2451	0.4085

Other includes timeslots 9 to 11:59 pm and 5 to 5:59 pm

Oi = total of all 7 days for the ith time slot.

CHECK: ΣOi = ΣEi. Done			k	2	s	0	α	0.05
χ2cal	0.4085	DF	2	χ2crit	5.9915

Decision

Since, χ2_cal > χ2_crit, H₀ is accepted.

Conclusion

Since H₀ is accepted, we conclude that manager’s instinct that 60% of the daily customers come between 6:00pm and 8:59pm and the remaining 40% of customers come at other times during the operating hours is validated. Answer

DONE