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 4.23 days 93% 91%
2008 4.22 days 98% 94%
2009 4.17 days 98% 96%
2010 4.10 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) 4.1800 0.0298 0.0595 4.10 4.1950 4.23
aspirin (%) 97.2500 93 98.0000 100
lvsd (%) 91 95.0000 98

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

Answer:

Descriptive Statistics for Aspirin (%)

Step 1: Calculate Standard Error

The formula for standard error (SE) is $SE = \frac{s}{\sqrt{n}}$, where $s$ is the sample standard deviation and $n = 4$ (number of years). First, find the sample standard deviation ($s$) of Aspirin data: 93, 98, 99, 100.
Mean ($\bar{x}$) of Aspirin: $\frac{93 + 98 + 99 + 100}{4} = 97.5$ (matches given mean).
Deviations from mean: $93 - 97.5 = -4.5$, $98 - 97.5 = 0.5$, $99 - 97.5 = 1.5$, $100 - 97.5 = 2.5$.
Squared deviations: $(-4.5)^2 = 20.25$, $(0.5)^2 = 0.25$, $(1.5)^2 = 2.25$, $(2.5)^2 = 6.25$.
Sum of squared deviations: $20.25 + 0.25 + 2.25 + 6.25 = 29$.
Sample variance ($s^2$): $\frac{29}{4 - 1} = \frac{29}{3} \approx 9.6667$.
Sample standard deviation ($s$): $\sqrt{\frac{29}{3}} \approx 3.1091$.
Standard Error: $\frac{3.1091}{\sqrt{4}} = \frac{3.1091}{2} \approx 1.5546$.

Step 2: Confirm Standard Deviation

Sample standard deviation $s = \sqrt{\frac{29}{3}} \approx 3.1091$.

Descriptive Statistics for LVSD (%)

Step 1: Calculate Mean

LVSD data: 91, 94, 96, 98.
Mean ($\bar{x}$): $\frac{91 + 94 + 96 + 98}{4} = \frac{379}{4} = 94.7500$.

Step 2: Calculate Standard Error

First, find sample standard deviation ($s$).
Deviations from mean: $91 - 94.75 = -3.75$, $94 - 94.75 = -0.75$, $96 - 94.75 = 1.25$, $98 - 94.75 = 3.25$.
Squared deviations: $(-3.75)^2 = 14.0625$, $(-0.75)^2 = 0.5625$, $(1.25)^2 = 1.5625$, $(3.25)^2 = 10.5625$.
Sum of squared deviations: $14.0625 + 0.5625 + 1.5625 + 10.5625 = 26.75$.
Sample variance ($s^2$): $\frac{26.75}{4 - 1} = \frac{26.75}{3} \approx 8.9167$.
Sample standard deviation ($s$): $\sqrt{\frac{26.75}{3}} \approx 2.9861$.
Standard Error: $\frac{2.9861}{\sqrt{4}} = \frac{2.9861}{2} \approx 1.4931$.

Step 3: Confirm Standard Deviation

Sample standard deviation $s = \sqrt{\frac{26.75}{3}} \approx 2.9861$.

Correlations (Using Pearson’s Correlation Formula: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$)

Correlation between LOS and Aspirin:

LOS data: 4.23, 4.22, 4.17, 4.10 (mean = 4.18).
Aspirin data: 93, 98, 99, 100 (mean = 97.5).
Calculate $\sum (x_i - \bar{x})(y_i - \bar{y})$:
$(4.23 - 4.18)(93 - 97.5) + (4.22 - 4.18)(98 - 97.5) + (4.17 - 4.18)(99 - 97.5) + (4.10 - 4.18)(100 - 97.5)$
$= (0.05)(-4.5) + (0.04)(0.5) + (-0.01)(1.5) + (-0.08)(2.5)$
$= -0.225 + 0.02 - 0.015 - 0.2$
$= -0.42$.

$\sum (x_i - \bar{x})^2$ (LOS variance numerator):
$(0.05)^2 + (0.04)^2 + (-0.01)^2 + (-0.08)^2 = 0.0025 + 0.0016 + 0.0001 + 0.0064 = 0.0106$.

$\sum (y_i - \bar{y})^2$ (Aspirin variance numerator): 29 (from earlier).

$r = \frac{-0.42}{\sqrt{0.0106 \times 29}} \approx \frac{-0.42}{\sqrt{0.3074}} \approx \frac{-0.42}{0.5544} \approx -0.757$.

Correlation between LOS and LVSD:

LVSD data: 91, 94, 96, 98 (mean = 94.75).
Calculate $\sum (x_i - \bar{x})(y_i - \bar{y})$:
$(4.23 - 4.18)(91 - 94.75) + (4.22 - 4.18)(94 - 94.75) + (4.17 - 4.18)(96 - 94.75) + (4.10 - 4.18)(98 - 94.75)$
$= (0.05)(-3.75) + (0.04)(-0.75) + (-0.01)(1.25) + (-0.08)(3.25)$
$= -0.1875 - 0.03 - 0.0125 - 0.26$
$= -0.49$.

$\sum (y_i - \bar{y})^2$ (LVSD variance numerator): 26.75 (from earlier).

$r = \frac{-0.49}{\sqrt{0.0106 \times 26.75}} \approx \frac{-0.49}{\sqrt{0.28355}} \approx \frac{-0.49}{0.5325} \approx -0.920$.

Correlation between Aspirin and LVSD:

$\sum (x_i - \bar{x})(y_i - \bar{y})$ (Aspirin - LVSD):
$(93 - 97.5)(91 - 94.75) + (98 - 97.5)(94 - 94.75) + (99 - 97.5)(96 - 94.75) + (100 - 97.5)(98 - 94.75)$
$= (-4.5)(-3.75) + (0.5)(-0.75) + (1.5)(1.25) + (2.5)(3.25)$
$= 16.875 - 0.375 + 1.875 + 8.125$
$= 26.5$.

$r = \frac{26.5}{\sqrt{29 \times 26.75}} \approx \frac{26.5}{\sqrt{775.75}} \approx \frac{26.5}{27.85} \approx 0.951$.

Final Descriptive Statistics (Filled):

• Aspirin (%):

Mean: 97.2500, Standard Error: 1.5546, Standard Deviation: 3.1091, Minimum: 93, Median: 98.0000, Maximum: 100.

• LVSD (%):

Mean: 94.7500, Standard Error: 1.4931, Standard Deviation: 2.9861, Minimum: 91, Median: 95.0000, Maximum: 98.

Final Correlations (Rounded to 3 decimals):

• LOS & Aspirin: $r \approx -0.757$
• LOS & LVSD: $r \approx -0.920$
• Aspirin & LVSD: $r \approx 0.951$

Example Filled Table for Descriptive Stats (Aspirin/LVSD):

Variable	Mean	Standard Error	Standard Deviation	Minimum	Median	Maximum
LVSD (%)	94.7500	1.4931	2.9861	91	95.0000	98

Example Filled Correlation Table:

	LOS	Aspirin
LVSD	-0.920	0.951

(Note: The correlation matrix is symmetric, so Aspirin-LVSD is 0.951, and diagonal is 1.000.)