TABLE 1
The general manager of a chain of pharmaceutical stores reported the results of a regression analysis, designed to predict the annual sales for all the stores in the chain (Y) – measured in millions of dollars. One independent variable used to predict annual sales of stores is the size of the store (X) – measured in thousands of square feet. Data for 14 stores were used to fit a linear model.
The results of the simple linear regression are provided below.
^
Y = 0.964 + 1.670X; SYX =$0.9664 million; 2 – tailed p value = 0.00004 (for testing ß1);
Sb1=0.157; X = 2.9124; SSX=Σ( Xi –X )2=37.924; n=14 ;
Referring to Table 1, the general manager wanted to test the null hypothesis that the true slope of size of store and annul sales was equal to one. The value of the test statistic is:
0.1570
0.4268
4.2675
1.6700
0.4168
.4268
To test the null hypothesis that the true slope of size of store and annual sales is equal to one, we need to calculate the test statistic.
The formula for the test statistic is given by:
t = (b1 - β1) / Sb1
Where:
b1 is the estimated slope coefficient (1.670)
β1 is the hypothesized value of the slope (1)
Sb1 is the standard error of the slope coefficient (0.157)
Substituting the values into the formula, we get:
t = (1.670 - 1) / 0.157
Calculating this equation, we get:
t ≈ 0.4268
Therefore, the value of the test statistic is approximately 0.4268.
To test the null hypothesis that the true slope of size of store and annual sales is equal to one, we need to calculate the test statistic. The test statistic is given by:
𝑡 = (𝑏₁ - 𝑛𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑠𝑙𝑜𝑝𝑒)/𝑆ℎ𝑜𝑟𝑡 𝑜𝑓 𝑏₁
where 𝑏₁ is the estimated slope coefficient, 𝑛𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑠𝑙𝑜𝑝𝑒 is the hypothesized slope (in this case, 1), and 𝑆ℎ𝑜𝑟𝑡 𝑜𝑓 𝑏₁ is the standard error of the estimated slope coefficient.
From the given information, we have 𝑏₁ = 1.670, 𝑛 = 14, 𝑆𝑏₁ = 0.157.
Plugging these values into the formula, we get:
𝑡 = (1.670 - 1) / 0.157
Simplifying further, we have:
𝑡 = 0.67 / 0.157
Calculating this division gives us:
𝑡 ≈ 4.2675
Therefore, the value of the test statistic is approximately 4.2675.