Rehabilitation Medicine and Chiropractic

发布时间：2024-06-28

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Rehabilitation Medicine and Chiropractic

Essential concepts in Statistics

Exercises

1. The following characteristics were noted for herpes patients in a therapy study:

Variable	Type
	nominal	ordinal	metric discrete	metric continuous
Age
Gender
No. of days since start of disease
Sensation of pain (no, mild, medium , strong)
No. of vesicles
Creatinine level

Indicate the correct level of measurement for the 6 characteristics.

2. The age of 7 patients in years: 27, 39, 40, 33, 30, 28, 34

a) Calculate the following statistical estimates: mean, median, range, standard deviation.

b) How does the median of these seven values change if the oldest patient is not 40, but 47 years old?

c) Draw the empirical distribution function.

3. The following boxplot describes the distribution of IPSS-Scores from 89 patients. Indicate which of the following statistical parameters can be read from the boxplot and, if possible, give the corresponding value.

Parameter

yes

value

Mean value
Median
upper quartile
Lower quartile
Minimum
Maximum
Standard deviation
Variance
IQR
Range
95% quantile

4. The age distribution of 802 passengers boarding the Titanic in Queenstown is shown in the figure as an empirical distribution function. Complete the following table by reading the values of the statistical parameters given there from the figure and specifying the correct type of measure.

Parameter

value (years)

location measure

variation measure

Median
lower quartile
upper quartile
Minimum
Maximum
IQR
Span

5. In a clinic, the proportion of patients suffering from diabetes mellitus is 20%. 30% of the patients suffer from heart disease, 5% have both diseases. What is the proportion of patients who have only one of the two diseases?

6. Ignaz Semmelweis determined for one month in 1846 that 24% of women giving birth in a ward of the Vienna maternity hospital had puerperal fever. At that time, the probability of a sick woman dying of puerperal fever was 80%˙ . What was the probability of a woman falling ill with puerperal fever and dying from it?

7. A total of 2805 new cases were reported to the clinical tumor registry in Nanjing in 2023. The places of residence of patients from Jiangsu Province had the following distribution:

Counties in Jiangsu (outside Nanjing) 1254

City of Nanjing 1258

Other districts in Jiangsu 176

What is the probability that a randomly selected registration form does not originate from Jiangsu?

8. In a screening examination, 15% of the people examined had heart disease and 10% had lung disease. 80% had neither disease. What was the proportion of people examined who had both heart and lung disease?

9. A health insurance company determines that only 8%˙ of car drivers who were wearing seat belts had head injuries in road accidents. In the case of drivers not wearing seatbelts, 62%˙ suffered no head injuries in an accident.

Despite the obligation to wear a seatbelt, 10% of all drivers still do not wear a seatbelt.

What is the probability that a driver after an accident with head injuries was not wearing a seatbelt?

10. You have just returned from vacation. During your stay, you learned that there is a rare viral disease there. You decide to have a test carried out after your return, as the chances of recovery are significantly better with early detection than after the outbreak of the disease. A few days after the test, your doctor calls you and tells you that your test is positive. He gave you additionally the following information:

1. In infected persons, tropical fever is detected in 99% of cases.

On the other hand, 98% of non-infected people are recognized as healthy.

2. Tropical fever only occurs in about one in every thousand tourists who visit this country

The disease shows no symptoms within the first few days. What is the probability that you are actually infected?

11. A study to assess the diagnostic significance of gallbladder ultrasonography for the detection of gallstones produced the following results:

Diagnosis after operation

Sum

gallstones present

gallstones

sonographic findings

gallstones present

410

468

gallstones

386

146

532

total

796

204

1000

Calculate sensitivity, specificity and predictive values.

12. In a screening examination of 8000 women, 500 of them were diagnosed with the disease. Subsequent special diagnostics confirmed 400 of these women as really ill and a further 120 cases of the disease were identified. Calculate the predictive values, the sensitivity and the specificity.

13. The sensitivity and specificity of some tests can be influenced. Which statements indicate a high sensi- tivity?

1 It is a disease with serious consequences for the patient.

2 There is a promising therapy.

3 The therapy may have serious side effects.

4 The therapy is very expensive.

5 The therapy places enormous psychological stress on patients.

6 False-positive findings can be clarified relatively easily.

14. In a cardiologic study, the question of whether the occurrence of new lesions in the coronary vessels is associated with the factor ‘“smoking“ was investigated. The following results were found in 230 patients after 3 years of observation (the absolute frequencies are shown):

	New lesions
	ja	nein

Smoker	34	33
Non-smoker	42	121

What are the relative risk and the odds ratio for the occurrence of new lesions?

15. The question to be investigated is whether the occurrence of complications in the mother is associated with a pregnancy entered as a risk in the maternity record. The result is as follows, with the absolute frequencies shown:

	Complications during birth
	yes	no

Risk Pregnancy	yes	33	58
Risk Pregnancy	no	22	84

Calculate the relative risk and the odds ratio.

16. Data on newborns is collected in a neonatal study. It is shown that the head circumference of the newborns is approximately normally distributed with a mean value of 34.8 cm and a standard deviation of 1.57 cm.

In which range are 95% of the values to be expected?

17. According to the manufacturer, certain tablets contain an average of 400 mg of an active ingredient. The active substance is normally distributed. The standard deviation is 5 mg. Calculate the proportion of tablets whose active ingredient content is greater than 395 mg and less than 405 mg.

18. It can be assumed that the hematocrit value of men is normally distributed with the parameters µ = 0.46 and σ = 0.03. The reference range for women is known and is given as 0.35-0.47. What is the probability that a randomly selected male person has a hematocrit value that is greater than the upper bound of the reference interval for women?

19. In a study to investigate the change in blood pressure in 32 hypertensive patients during the course of treatment, diastolic blood pressure is measured before and after treatment. The mean value of the differences (pre-value - post-value) is 21.7 mmHg and their standard deviation is 9.7 mmHg. State the 95% confidence interval for the expected value of the differences in the population.

(Note: The 97.5% quantile of the t-distribution with 31 degrees of freedom is 2.04)

20. The mean uric acid value in serum for men is given by a reference laboratory as 305 µmol/l. The analysis of 16 blood samples from women yielded an average value of ¯(x) = 260 µmol/l with a standard deviation of s = 50 µmol/l. Calculate the 95% confidence interval for the expected value of uric acid in serum in women.

(Note: The 0.975 quantile of the t-distribution with 15 degrees of freedom is 2.13)