Question

A hospital would like to develop a regression model to predict the total hospital bill for a patient based on his or her length of stay, number of days in the hospitais intensive care una (CU), and age of the patient Data for these variables can be found in the accompanying table Complete parts (a) through (e) below. Click the icon to view the data table a) Using technology, construct a regression model using all three independent variables, where y Cost, Age, XDays, and X-CU. (Round to the nearest Integer as needed.) b) Test the significance of each independent variable using a = 0.05. Identify the general form of the null and alternative hypotheses, where is the population regression coefficient for the thindependent variable HAVO HAVO Find the test statistic for the coefficient of each independent variable Variable (Hound to two decimal places as needed)

Answer 1

a)

Answer:

$\widehat{Y}=-705+116\times x_1+1253 \times x_2+2144\times x_3$

Explanation:

The regression equation is defined as,

$\widehat{Y}=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_3$

Now, the regression analysis is done in excel by following steps

Step 1: Write the data values in excel. The screenshot is shown below,

A B 1 Cost, y Age, x1 1457 20 1954! 2 Regression C D Days, x2 ICU, x3 1 -- 0 39 ? X Input Input y Range: Гок $A$1:5A548 Cancel Input X Range: $B$1:$D$48 Help Labels Confidence Level: Constant is Zero 95 % SF$1 Output options O Output Range: O New Worksheet Ply: O New Workbook Residuals Residuals Standardized Residuals Residual Plots Line Fit Plots Normal Probability Normal Probability Plots

Step 2: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

DATA REVIEW Data Analysis ? X Гок HE INSERT PAGE LAYOUT FORMULAS AL ZA X - Data Analysis Analysis Tools Fourier Analysis Histogram Moving Average Random Number Generation Rank and Percentile Regression Sampling t-Test: Paired Two Sample for Means t-Test: Two-Sample Assuming Equal Variances t-Test: Two-Sample Assuming Unequal Variances Cancel Help 36.6

Step 3: Select Input Y Range: 'Y' column, Input X Range: 'x1, x2 and x3' column then OK. The screenshot is shown below,

C D Days, x2 ICU, x3 1 A B 1 Cost, y Age, x1 14571 20 1954 39 2721! 34 Regression ? X Input Input Y Range: Гок SAS1:5A548 Cancel Input X Range: $B$1:$D$48 Help Labels Confidence Level: Constant is Zero 95 % SF51 Output options O Qutput Range: O New Worksheet Ply: O New Workbook Residuals Residuals Standardized Residuals Residual Plots Line Fit Plots Normal Probability Normal Probability Plots

The result is obtained. The screenshot is shown below,

1 SUMMARY OUTPUT Regression Statistics 4 Multiple R 0.871862 5 R Square 0.760144 6 Adjusted R Square 0.74341 7 Standard Error 4039.554 8 Observations 47 10 ANOVA df 12 Regression 13 Residual 14 Total SSMSF gnificance F 3 2.22E+09 7.41E+08 45.42477 2.18E-13 43 7.02E+08 16317993 46 2.93E+09 17 Intercept 18 Age, x1 19 Days, x2 20 ICU, x3 Coefficientandard Err Stat P-value Lower 95%Upper 95% -704.936 1740.982 -0.40491 0.687553 -4215.96 2806.089 115.9269 38.69164 2.996174 0.004525 37.89775 193.956 1253.062 278.5935 4.497817 5.14E-05 691.2251 1814.9 2144.285 1199.377 1.787831 0.080852 -274.49 4563.06

The regression equation is,

Cost = -705 + 116 x Age + 1253 x Days + 2144 x ICU

b)

The null and alternative hypotheses are defined as,

Ho : Slope, B; = 0

Hp : Slope, B; + 0

From the regression output summary,

t-statistic

Variable	t-statistic
x1	3.00
x2	4.50
x3	1.79

P-value

Variable	P-value
x1	0.005
x2	0.000
x3	0.081

Interpretation of p-value

For x1

Since the p-value associated with x1 is less than , reject the null hypothesis. There is sufficient evidence to conclude that there is a relationship between x1 and y.

= 0,05

For x2

Since the p-value associated with x1 is less than , reject the null hypothesis. There is sufficient evidence to conclude that there is a relationship between x2 and y.

= 0,05

For x3

Since the p-value associated with x1 is greater than , do not reject the null hypothesis. There is not sufficient evidence to conclude that there is a relationship between x3 and y.

= 0,05

c)

Answer: A.

Explanation: Since p-value is less than for variable x1 and x2, these two variable have a significant relationship with y.

= 0,05

d)

From the regression output summary,

The 95% confidence interval is from -274 to 4563

Correct option: A.

Explanation: Since the variable x3 is the number of days in the ICU, the 95% confidence interval of the slope coefficient can be interpreted as we 95% confident that for 1 extra day stay in ICU, the average total hospital bill will lie within this limit.

e)

Since the confidence interval includes 0, there is not sufficient evidence to conclude that the true population coefficient $\beta_3$, is not 0 and that there is a relationship between the number of days spent in the ICU and the total hospital bill.