bookings <- read.table("../data/first_bookings_datetime_sample.txt",
                       header=TRUE) # loading the data
str(bookings$datetime) # understanding what data type we are dealing with

Our data is contained in a character vector, which is not ideal for performing statistical operations. Therefore, we will convert the content of datetime to a POSIXlt date-time format and extract the components that interest us.

Calculate the time of each booking in minutes since the start of the day. Store the result in a new variable - minday

Since we know that EZTABLE would like to start a promotion for new members to make their bookings earlier in the day, we focus on extracting the hours and minutes after converting datetime to a POSIXlt object. We do not focus on date.

# Converting datetime to a POSIXlt object, extracting hours and minutes
hours <- as.POSIXlt(bookings$datetime, format="%m/%d/%Y %H:%M")$hour 
mins <- as.POSIXlt(bookings$datetime, format="%m/%d/%Y %H:%M")$min

Afterwards, we calculate the time of each booking in minutes since the start of the day. So, we multiply the hours by 60 and add the minutes, resulting in a single numerical value that represents the minute of the day:

minday <- hours * 60 + mins

Plot the density of minday

Now, we shall see the density of minday so as to get an idea of how the values of this vector are distributed:

plot(density(minday), main="Minute of the day of first ever booking",
     ylab = "", xlab = "Minutes", col="cornflowerblue", lwd=2)

We see a distribution that is similar to a bimodal, with a peek of booking times around 700 minutes and another around 1100 minutes, which likely represent booking requests before lunch and dinner time. Next, we are going to conduct an analysis of the mean booking time.

2. Mean Booking Times - Bootstrapping

Calculate the mean booking time and its standard error

We shall start by computing the mean booking time and its standard error. The standard error is computed by dividing the standard deviation of the booking time by the square root of the total number of bookings:

minday_mean <- mean(minday) # mean
minday_mean

minday_se <- sd(minday)/sqrt(length(minday)) # standard error
minday_se

The mean booking time corresponds to 942.4964 minutes. To convert the time back to a familiar format, we use a custom function, located in the functions.R script:

source(file = "../R/functions.R")
mins_to_time(minday_mean) # custom function that converts number of minutes to time of the day

Develop a 95% confidence interval for the mean booking time

Recall that the data we have available is only a sample; now that we know mean and standard error, we are able to build a confidence interval and get a clearer idea about the true average booking time:

# 95% CI:
lower_bound <- minday_mean - 1.96 * minday_se
lower_bound

upper_bound <- minday_mean + 1.96 * minday_se
upper_bound

The interval we just built suggests that we can be 95% confident that the true average booking time is between 941.2308 and 943.2308 minutes, or that it is included between 15:41 and 15:44.

mins_to_time(lower_bound)

mins_to_time(upper_bound)

Implement a bootstrap approach to generate 2000 new samples from the original dataset

Next, we are going to employ bootstrapping. This is a technique that resamples a dataset to create several simulated samples. To implement bootstrapping, we use the replicate() function, paired with sample(). Let’s generate 2000 new samples from the original sample data:

# Setting seed
set.seed(100)

# Creating 2000 new samples (bootstrapped samples) from the starting data
new_samples <- replicate(n = 2000,
                         sample(minday, length(minday), replace = TRUE))

Visualize the distribution of the means from the bootstrapped samples

We might be interested in seeing how the 2000 bootstrapped samples look like in comparison with the starting sample. We can display them on the same plot. We achieve this by applying the custom function plot_resamples(), included in the functions.R script. This function plots the density of a sample and then applies a function on the sample; if paired with apply(), it allows us to plot each bootstraped samples’ density and compute their means.

# Create an empty plotting space
plot(density(minday), lwd=0, ylim=c(0, 0.004), xlab = "Minutes", ylab = "",
     main = "Original sample vs. Bootstrapped samples with Mean indicators")

# Plot and get means of all bootstrapped samples
sample_means <- apply(new_samples, 2,
                      FUN = function(x) plot_resamples(x, mean))

# Add starting sample density to the plot
lines(density(minday), lwd = 2)

# add vertical lines for the means of each the original sample and each new sample 
abline(v = sample_means, col = rgb(0.0, 0.4, 0.0, 0.05))
abline(v = minday_mean, lwd = 1, col = "red")

We notice that the bootstrapped samples’ densities are all very close to the original sample. The same goes for the bootstrapped means, the confidence interval is certainly very narrow. Now, let’s visualize the density of the 2000 bootstrapped means. We can take a look at how they compare with the original mean value from the starting sample.

plot(density(sample_means), lwd = 2, col = "cornflowerblue",
     main = "Density of Bootstrapped samples with Original mean indicator",
     ylab = "", xlab = "Minutes")
abline(v = minday_mean, lwd = 2, col = "tomato") # adding vertical line for original sample mean

We see that the bootstrapped means do not vary largely from the mean of the original sample. Most values are included between 941 and 944 minutes, with only a few means being outside of this interval.

Calculate and interpret the 95% confidence interval of the bootstrapped means

We know what the 95% confidence interval is for the original sample’s mean, now we can build a new interval for the resampled means. We can build this interval using the quantile() function:

resampled_means_CI <- quantile(sample_means, probs = c(0.025, 0.975))
resampled_means_CI

The 95% CI of the bootstrapped means ranges from 941.2731 to 943.766 (or from 15:41 to 15:44), which is very similar to the CI of the original sample’s mean.

mins_to_time(resampled_means_CI)

This means that we are 95% confident that the true average booking time falls between 15:41 and 15:44. With this confidence interval, we complete the analysis of the average booking time.

3. Median Booking Times - Bootstrapping

Determine the median booking time

Now, we focus on the median booking time, in other words, the time of the day by which half of the customers has made their first booking.

# Computing median booking time
minday_median <- median(minday)
minday_median

# Converting minutes to time
mins_to_time(minday_median)

We see that the median booking time is 1040 minutes, in other words, it’s 17:20.

Visualize the distribution of the medians from the bootstrapped samples

We can display the bootstrapped medians and compare them with the original sample’s median booking time. We shall use plot_resamples() again.

# Create an empty plotting space
plot(density(minday), lwd=0, ylim=c(0, 0.004), xlab = "Minutes", ylab = "",
     main = "Original sample vs. Bootstrapped samples with Median indicators")

# Plot and get means of all bootstrapped samples
sample_medians <- apply(new_samples, 2,
                        FUN = function(x) plot_resamples(x, median))

# Add starting sample density to the plot
lines(density(minday), lwd = 2)

# add vertical lines for the means of each the original sample and each new sample 
abline(v = sample_medians, col = rgb(0.0, 0.4, 0.0, 0.05))
abline(v = minday_median, lwd = 1, col = "red")

Interestingly, we see that the bootstrapped medians are close to the original sample’s median, however, there are gaps in between these values. This is due to the fact that all the bootstrapped median values are represented by one of these discrete values:

sort(unique(sample_medians))

plot(density(sample_medians), lwd = 2, col = "cornflowerblue",
     main = "Density of Bootstrapped samples with Original median indicator",
     ylab = "", xlab = "Minutes")
abline(v = minday_median, lwd = 2, col = "tomato") # add vertical line for original sample median

As we can see, the bootstrapped medians are distributed quite widely and non-normally. This makes it hard to use them for inference.

Estimate and interpret the 95% confidence interval of the bootstrapped medians

resampled_median_CI <- quantile(sample_medians, probs = c(0.025, 0.975))
resampled_median_CI

The confidence interval shows that we can be 95% confident that the true median booking time is included between minutes 1020 and 1050, else:

mins_to_time(resampled_median_CI)

Between 17:00 and 17:30. This is a window of 30 minutes, which is quite large. The calculation of the confidence interval for the median concludes the median analysis.

4. Business Implications

Insights

The bimodal distribution seen at the end of the data exploration section suggests two peak periods, likely corresponding to lunch and dinner times. This pattern indicates that new members to EZTABLE prefer these periods for their first bookings. This trend presents an opportunity to shift some of this demand to earlier times. For example, offering special promotions or incentives for bookings made before the usual lunch peak could help redistribute the customer flow. The narrow confidence interval for the mean booking time seen in the mean analysis section suggests a consistent pattern in customer behavior. Since the mean is later in the day (15:42), EZTABLE can design targeted promotions aimed at times just before the average, gradually encouraging customers to book earlier. On the other hand, the median analysis showed us a distribution that doesn’t allow us to conduct inference easily.

Business Strategy Recommendations

Considering what we have discovered so far, EZTABLE could employ the following strategies in order to encourage early bookings:

These are a few simple recommendations for EZTABLE, which are designed to leverage the insights from the booking time analysis to effectively encourage early-day bookings, thereby aiding EZTABLE in optimizing restaurant reservations and enhancing customer flow.

Restaurant Booking Analysis with Bootstrapping

1. Data Exploration

Load the data and convert datetime into a POSIXlt date-time format

Calculate the time of each booking in minutes since the start of the day. Store the result in a new variable - minday

Plot the density of minday