Question

A department store manager wants to estimate the number of customers that enter the store from noon until closing at 9 P.M. The table shows the number of customers \(N\) entering the store during a randomly selected minute each hour from \(t-1\) to \(t,\) with \(t=0\) corresponding to noon. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{N} & 6 & 7 & 9 & 12 & 15 & 14 & 11 & 7 & 2 \\ \hline \end{array} $$ (a) Draw a histogram of the data. (b) Estimate the total number of customers entering the store between noon and 9 P.M. (c) Use the regression capabilities of a graphing utility to find a model of the form \(N(t)=a t^{3}+b t^{2}+c t+d\) for the data. (d) Use a graphing utility to plot the data and graph the model. (e) Use a graphing utility to evaluate \(\int_{0}^{9} N(t) d t,\) and use the result to estimate the number of customers entering the store between noon and 9 P.M. Compare this with your answer in part (b). (f) Estimate the average number of customers entering the store per minute between 3 P.M. and 7 P.M.

Short answer

The short answer will depend on the specific dataset, but it will involve the sums and averages of the numbers in the table, the specific model obtained from regression, and the evaluation of the integral of the model.

Step-by-step solution

Drawing the Histogram

For each hour from 1 to 9 (representing the hours after noon), create a bar graph with the number of customers entering the store as the height of each bar. The histogram can be created using graphing software or drawn manually.

Estimating the Total Number of Customers

Estimate the total number of customers by adding up all the numbers given in the table containing the number of customers per minute.

Finding the Model

Use a graphing utility or statistical software to perform a cubic regression on the data points. The result will be a third degree polynomial model of the form \(N(t) = a t^3 + b t^2 + c t + d\). The constants a, b, c, and d depend on the specific dataset.

Plotting the Data and Model

Use a graphing utility to plot the data points and the model obtained from the regression. This will help visualize how well the model fits the data.

Evaluating the Integral

Use a graphing utility to evaluate the integral \(\int_0^9 N(t) dt\). This integral represents the estimated total number of customers entering the store between noon and 9 P.M.

Estimating Average Number of Customers

To estimate the average number of customers entering the store per minute between 3 P.M. and 7 P.M., take the values at 3 P.M., 4 P.M., 5 P.M., 6 P.M., and 7 P.M., add them and divide by 5.

Key concepts

Cubic Regression

Cubic regression is an analytical tool used in statistics to fit a cubic equation to a set of data points. For our department store manager wanting to predict customer entry patterns, this tool offers a mathematical representation of the trend. To perform cubic regression, one would use the form:
\( N(t) = a t^3 + b t^2 + c t + d \).
The letters \( a \), \( b \), \( c \), and \( d \) represent the coefficients that will be calculated through the regression analysis using the dataset provided. With graphing or statistical software, these coefficients are established to minimize the difference between the actual data values and the values predicted by the model.

The advantage of using a cubic model over a simple linear one is its ability to capture more complex behaviors, such as the non-linear ebb and flow in customer arrivals throughout the day. In a business context, understanding these patterns can be vital for managerial decisions, such as staff scheduling and stock preparation.

Histogram Data Representation

A histogram offers a straightforward way to visualize data that helps spot trends and patterns. In the context of the department store, a histogram can represent customer entries at different times of the day.

To create a histogram for our given exercise, each hour is marked along the horizontal axis, and the vertical axis corresponds to the number of customers encountered in the sample minute. Thus, bars of varying heights are formed, illustrating the flow of traffic into the store. The areas of these bars indicate volume, allowing for quick assessments.

• Higher bars show peak hours with more customers.
• Shorter bars indicate off-peak times with fewer customers.

The visual clarity that histograms provide is particularly useful for those in business when identifying rush hours and quieter periods, enabling effective strategic planning.

Definite Integral Estimation

In calculus, the definite integral is a fundamental concept used to calculate the accumulation of quantities, such as estimating total number of customer visits in our department store scenario. The integral
\( \[ \int_{0}^{9} N(t) dt \] \)
represents the total customer entries from noon (\( t = 0 \)) until closing (\( t = 9 \)) P.M., when \( N(t) \) is given by the cubic model from our regression analysis.

By evaluating the definite integral using graphing utilities, the manager can compare this estimate to the one obtained simply by adding up individual counts. While simple addition can give a straightforward estimate, the integral considers the continuous model over time, potentially offering a more refined approximation, especially useful in making projections for future business planning. The integral's result captures the entire pattern of customer entries, unlike discrete counts that may miss in-between fluctuations.