The Art of Mathematics in Business. Dr Jae K Shim
labor, and so on, thereby causing potentially huge financial losses.
Business people need to keep track of past prediction records to ensure that (1) future costs can be minimized and (2) better forecasting methods can be developed.
26. χ2 (Chi-Square Test)
Introduction
The χ2 (chi-square) test is a statistical test of the significance of a difference between classifications or sub-classifications. The test is applied to sample data in testing whether or not two qualitative population variables are independent.
How is the test performed?
The χ2 test involves three steps.
Step 1: Calculate the χ2 statistic, which is defined as:
Where f0 = individual observed frequencies of each class
fe = individual expected frequencies of each class
Step 2. Find the table value at a given level of significance (see Table 6 in the Appendix).
Step 3. If the calculated value is greater than the table value, we reject the full hypothesis, which means that the two variable are classifications are associated.
Example
Consider the following survey regarding restaurants:
The null hypothesis is: The menu and the indoor/outdoor cafes are independent.
In order to calculate χ2, we need to construct an expected value table on the basis of the assumption that menu and indoor/outdoor cafes are independent of each other. If no association exists, it is to be expected that the proportion of à la carte and prix fixe restaurants with outdoor cafes or tables will be the same as that without outdoor cafes. First, we compute the expected frequencies based on the premise of independence:
635/844 = 0.7524209/844 = 0.2476
Now, we can compute the expected values from the proportions of totals as follows:
0.7524×646=486
0.2476×646=160
0.7524×198=149
0.2476×198=049
These expected values give the following table:
Step 1. Calculate χ2. We are interested in how far the observed table differs from the expected table.
=14.171
Step 2. χ2 value at the 0.05 level of significance with one degree of freedom (from Table 6 in the Appendix) is 3.841. The degree of freedom is calculated as (no. rows − 1) × (no. of columns – 1) = (2 – 1) × (2 – 1) = 1.
Step 3. As shown in Fig. 26.1, since the calculated value is greater than the table value (14.171 > 3.841), we reject the null hypothesis, which means that the menu is associated with the outdoor/indoor restaurant setup.
Figure 26.1: Chi-square test and rejection area.
How is it used and applied?
The chi-square test has many applications in business. It is a statistical test of independence (or association), to determine if membership in categories of one variable is different as a function of membership in the categories of a second variable. It is important to note, however, that there are limitations to this test: (1) The sample must be big enough for the expected frequencies for each cell (rule of thumb: at least 5); and (2) the test does not tell anything about the direction of an association.
Managers need to know whether the differences they observe among several sample proportions are significant or due only to chance. For example, marketing managers are concerned that their brand’s share may be unevenly distributed throughout the country. They conduct a survey in which the country is divided into specific number of geographic regions and see if consumer’s decisions as to whether or not to purchase the company’s brand has anything to do with the geographic location. As another example, a financial manager might be interested in the differences in capital structure within different firm sizes in a certain industry. To see if firm sizes have the same capital structure (or firm sizes have nothing to do with the capital structure), what he or she needs to do is to survey a group of firms with assets of different amounts and divide them into groups. Each firm can be classified according to predetermined debt/equity ratio groups.
27. Cash Flow Statement
Introduction
Current profitability is only one important factor in success. Also essential is cash flow. In fact, a business can be profitable and still have a cash crisis. An example is a small business with a high level of credit sales but with a very long collection period. The business shows a profit but does not have the cash from those sales.
It is essential to know your cash flow in order to plan adequately. Should you cut back on cash payments? Where are you obtaining cash flow? Where are you spending your money? What products are cash drains or cash surpluses? Is there adequate money to pay bills and purchase required equipment? Are you liquid?
A statement of cash flows is useful because it provides valuable data that are not available in the balance sheet and income statement. The statement presents the sources and uses of cash and is a basis for cash flow analysis.
How is it Computed?
The statement of cash flows classifies cash receipts and cash payments from (1) operating, (2) investing, and (3) financing activities. Let’s look at each of these major sections.
1.Operating activities relate to the manufacturing and selling of or the rendering of services. Cash inflows that come from operating activities include (a) cash sales or collections on receivables and (b) cash receipts from interest income and dividend income. Cash outflows include (a) cash paid for merchandise and (b) cash paid for expenses.
2.Investing activities relate to the purchase and sale of fixed assets (such as equipment and machinery) and the purchase of stocks and bonds of other businesses. Cash inflows comprise (a) amounts received from selling fixed assets and (b) receipts from sales of stocks and bonds of other companies. Cash outflows include (a) payments to purchase fixed assets and (b) disbursements to buy stocks and bonds of other entities.
3.Financing activities relate to borrowing and repayment and to issuing stock and reacquiring previously issued shares. Cash inflows comprise (a) funds obtained from loans and (b) funds received from the sale of stock. Cash outflows include (a) paying off debt, and (b) repurchase of stock.
Example
Consider Mr. Paul, who owns a business that sells appliances. The statement of cash flows for this small business follows: