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Q1. Ages for ID 1-10
Ages: 68, 50, 58, 58, 47, 51, 51, 40, 40, 69
[
\text{Mean age}=\frac{532}{10}=53.2\text{ years}
]
Sorted ages: 40, 40, 47, 50, 51, 51, 58, 58, 68, 69
[
\text{Median}=\frac{51+51}{2}=51\text{ years}
]
Answer: Mean = 53.2 years; Median = 51 years.
Q2. ID 11-20
(a) Range of survival months
Survival months: 64, 92, 56, 38, 64, 49, 105, 62, 107, 77
[
\text{Range}=107-38=69\text{ months}
]
(b) Mean deviation of age from mean
Ages: 68, 46, 65, 48, 62, 61, 56, 43, 48, 60
[
\bar{x}=\frac{557}{10}=55.7
]
[
\text{Mean deviation}=\frac{\sum|x-\bar{x}|}{10}
=\frac{75.6}{10}=7.56
]
Answer: Range = 69 months; Mean deviation of age = 7.56 years.
Q3. Regional nodes examined for ID 21-30
Values: 15, 15, 4, 18, 26, 31, 25, 14, 14, 10
[
\bar{x}=\frac{172}{10}=17.2
]
[
\sum(x-\bar{x})^2=585.6
]
Using the sample formula:
[
s^2=\frac{585.6}{9}=65.07
]
[
s=\sqrt{65.07}=8.07
]
Answer: Sample variance = 65.07; Sample standard deviation = 8.07.
If population formulas are required: variance = 58.56, SD = 7.65.
Q4. Mortality by marital status, ID 1-15
| Marital category | Alive | Dead | Total | Proportion dead |
|---|---|---|---|---|
| Married | 9 | 1 | 10 | (1/10=10%) |
| Non-married: single/divorced/widowed | 4 | 1 | 5 | (1/5=20%) |
Answer:
- Married patients dead = 10%
- Non-married patients dead = 20%
Higher mortality is seen in the non-married group.
Q5. Chi-square test: Estrogen status vs survival status, ID 1-15
Observed table
| Estrogen status | Alive | Dead | Total |
|---|---|---|---|
| Positive | 12 | 2 | 14 |
| Negative | 1 | 0 | 1 |
| Total | 13 | 2 | 15 |
Expected frequencies
[
E_{\text{Positive, Alive}}=\frac{14\times13}{15}=12.13
]
[
E_{\text{Positive, Dead}}=\frac{14\times2}{15}=1.87
]
[
E_{\text{Negative, Alive}}=\frac{1\times13}{15}=0.87
]
[
E_{\text{Negative, Dead}}=\frac{1\times2}{15}=0.13
]
[
\chi^2=\sum\frac{(O-E)^2}{E}=0.165
]
Degrees of freedom:
[
df=(2-1)(2-1)=1
]
Critical value at (df=1), 5% significance level = 3.84
[
0.165 < 3.84
]
Answer: (\chi^2 = 0.165), (df=1). The association is not statistically significant. Estrogen status and alive/dead status are not significantly associated in this sample.
Note: several expected frequencies are below 5, so Fisher's exact test would be more suitable in practice.
Q6. Correlation and regression, ID 1-10
Let:
- (X) = Regional nodes examined
- (Y) = Regional nodes positive
[
\sum X=136,\quad \sum Y=49
]
[
\sum X^2=2348,\quad \sum Y^2=551,\quad \sum XY=865
]
[
r=\frac{n\sum XY-\sum X\sum Y}
{\sqrt{[n\sum X^2-(\sum X)^2][n\sum Y^2-(\sum Y)^2]}}
]
[
r=\frac{10(865)-136(49)}
{\sqrt{[10(2348)-136^2][10(551)-49^2]}}
]
[
r=0.505\text{ approximately}
]
So, there is a moderate positive correlation between regional nodes examined and regional nodes positive.
Regression equation predicting regional node positive from regional node examined
[
b=\frac{S_{XY}}{S_{XX}}=\frac{198.6}{498.4}=0.398
]
[
a=\bar{Y}-b\bar{X}=4.9-(0.398)(13.6)=-0.519
]
[
\boxed{\hat{Y}=-0.519+0.398X}
]
Where:
- (\hat{Y}) = predicted regional nodes positive
- (X) = regional nodes examined
Answer: Pearson correlation coefficient, (\boxed{r=0.505}).
Regression equation: (\boxed{\hat{Y}=-0.519+0.398X}).
Regression equation: (\boxed{\hat{Y}=-0.519+0.398X}).
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