Question

a key hospital outcome measure of clinical performance is length of stay (los), that is, the number of days a patient is hospitalized. for patients at one hospital with acute myocardial infarction (heart attack), the length of stay over the past four years has consistently decreased. the hospital also has data for various treatment options such as the percentage of patients who received aspirin upon arrival and cardiac medication for left ventricular systolic dysfunction (lvsd). the data are as follows:
year average los aspirin on arrival lvsd medication
2007 4.31 days 92% 91%
2008 4.31 days 96% 93%
2009 4.13 days 99% 96%
2010 4.30 days 100% 98%
calculate descriptive statistics for the data. do not round intermediate calculations. round your answers to four decimal places, when needed.
descriptive statistics: los, aspirin, lvsd
variable mean standard error standard deviation minimum median maximum
los (days)
aspirin (%)
lvsd (%)
calculate correlations for the data. do not round intermediate calculations. use a minus sign for negative values, if any. round your answers to three decimal places.
correlations: los, aspirin, lvsd
los aspirin
aspirin
lvsd
choose the correct graph showing the los as a function of the aspirin on arrival.
the correct graph is -select-

Explanation:

Step1: Calculate mean of LOS

Let \(x_1 = 4.31\), \(x_2=4.31\), \(x_3 = 4.13\), \(x_4=4.30\). Mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{4.31 + 4.31+4.13 + 4.30}{4}=\frac{17.05}{4}=4.2625\)

Step2: Calculate standard - error of LOS

First, calculate the variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\).
\((x_1-\bar{x})=(4.31 - 4.2625)=0.0475\), \((x_2-\bar{x})=(4.31 - 4.2625)=0.0475\), \((x_3-\bar{x})=(4.13 - 4.2625)=- 0.1325\), \((x_4-\bar{x})=(4.30 - 4.2625)=0.0375\)
\(\sum_{i = 1}^{4}(x_i-\bar{x})^2=(0.0475)^2+(0.0475)^2+(-0.1325)^2+(0.0375)^2\)
\(=0.00225625+0.00225625 + 0.01755625+0.00140625=0.023475\)
Variance \(s^{2}=\frac{0.023475}{3}\approx0.007825\)
Standard - deviation \(s=\sqrt{0.007825}\approx0.08846\)
Standard error \(SE=\frac{s}{\sqrt{n}}=\frac{0.08846}{\sqrt{4}}\approx0.04423\)

Step3: Find minimum, median and maximum of LOS

Arrange in ascending order: \(4.13,4.30,4.31,4.31\)
Minimum \(=4.13\), Median \(\frac{4.30 + 4.31}{2}=4.305\), Maximum \(=4.31\)

Step4: Calculate mean of Aspirin

Let \(y_1 = 92\), \(y_2=96\), \(y_3 = 99\), \(y_4=100\). Mean \(\bar{y}=\frac{92 + 96+99 + 100}{4}=\frac{387}{4}=96.75\)

Step5: Calculate standard - error of Aspirin

\((y_1-\bar{y})=(92 - 96.75)=-4.75\), \((y_2-\bar{y})=(96 - 96.75)=-0.75\), \((y_3-\bar{y})=(99 - 96.75)=2.25\), \((y_4-\bar{y})=(100 - 96.75)=3.25\)
\(\sum_{i = 1}^{4}(y_i-\bar{y})^2=(-4.75)^2+(-0.75)^2+(2.25)^2+(3.25)^2\)
\(=22.5625 + 0.5625+5.0625 + 10.5625=38.75\)
Variance \(s_y^{2}=\frac{38.75}{3}\approx12.9167\)
Standard - deviation \(s_y=\sqrt{12.9167}\approx3.5939\)
Standard error \(SE_y=\frac{s_y}{\sqrt{n}}=\frac{3.5939}{\sqrt{4}}\approx1.7970\)
Minimum \(=92\), Median \(\frac{96 + 99}{2}=97.5\), Maximum \(=100\)

Step6: Calculate mean of LVSD

Let \(z_1 = 91\), \(z_2=93\), \(z_3 = 96\), \(z_4=98\). Mean \(\bar{z}=\frac{91+93 + 96+98}{4}=\frac{378}{4}=94.5\)

Step7: Calculate standard - error of LVSD

\((z_1-\bar{z})=(91 - 94.5)=-3.5\), \((z_2-\bar{z})=(93 - 94.5)=-1.5\), \((z_3-\bar{z})=(96 - 94.5)=1.5\), \((z_4-\bar{z})=(98 - 94.5)=3.5\)
\(\sum_{i = 1}^{4}(z_i-\bar{z})^2=(-3.5)^2+(-1.5)^2+(1.5)^2+(3.5)^2\)
\(=12.25+2.25 + 2.25+12.25=29\)
Variance \(s_z^{2}=\frac{29}{3}\approx9.6667\)
Standard - deviation \(s_z=\sqrt{9.6667}\approx3.1091\)
Standard error \(SE_z=\frac{s_z}{\sqrt{n}}=\frac{3.1091}{\sqrt{4}}\approx1.5546\)
Minimum \(=91\), Median \(\frac{93 + 96}{2}=94.5\), Maximum \(=98\)

Step8: Calculate correlations

The correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}\)
\(\sum_{i = 1}^{4}(x_i-\bar{x})(y_i-\bar{y})=(0.0475)\times(-4.75)+(0.0475)\times(-0.75)+(-0.1325)\times(2.25)+(0.0375)\times(3.25)\)
\(=-0.225625-0.035625 - 0.298125+0.121875=-0.4375\)
\(r_{LOS - Aspirin}=\frac{-0.4375}{\sqrt{0.023475\times38.75}}\approx - 0.457\)
Similarly, \(\sum_{i = 1}^{4}(x_i-\bar{x})(z_i-\bar{z})=(0.0475)\times(-3.5)+(0.0475)\times(-1.5)+(-0.1325)\times(1.5)+(0.0375)\times(3.5)\)
\(=-0.16625-0.07125 - 0.19875+0.13125=-0.295\)
\(r_{LOS - LVSD}=\frac{-0.295}{\sqrt{0.023475\times29}}\approx - 0.357\)
\(\sum_{i = 1}^{4}(y_i-\bar{y})(z_i-\bar{z})=(-4.75)\times(-3.5)+(-0.75)\times(-1.5)+(2.25)\times(1.5)+(3.25)\times(3.5)\)
\(=16.625 + 1.125+3.375+11.375=32.5\)
\(r_{Aspirin - LVSD}=\frac{32.5}{\sqrt{38.75\times29}}\approx0.977\)

Answer:

Variable	Mean	Standard Error	Standard Deviation	Minimum	Median	Maximum
Aspirin (%)	96.7500	1.7970	3.5939	92.0000	97.5000	100.0000
LVSD (%)	94.5000	1.5546	3.1091	91.0000	94.5000	98.0000
Correlations	LOS	Aspirin	LVSD
Aspirin	-0.457
LVSD	-0.357	0.977