Biostatistics

Step 1: General properties of chi-square distribution.
The chi-square distribution is widely used in hypothesis testing, especially in tests of independence and goodness of fit.
It is defined only for non-negative values.
Step 2: Important characteristics.
- The chi-square distribution starts at 0 (it cannot take negative values).
- It is positively skewed and not symmetrical, though it approaches normal distribution as the degree of freedom increases.
- The mean of a chi-square distribution is equal to its degrees of freedom ( ).
- The variance of chi-square distribution is equal to .
Step 3: Check each statement.
- (A) Correct, chi-square starts at 0.
- (B) Incorrect, it is not symmetrical, it is skewed.
- (C) Correct, mean = df.
- (D) Incorrect, variance is not equal to df.
- (E) Correct, variance = 2 df.
Step 4: Conclusion.
Thus, the correct combination is A, C, and E.

Step 1: Define precision and bias.
- Precision: Consistency of repeated measurements (low variability).
- Bias: Deviation of the mean measurement from the true value.
Step 2: Instrument A.
Shows scattered values around the true level. Not precise, not unbiased. Hence statement (A) is incorrect.
Step 3: Instrument B.
Values are tightly clustered (precise), but consistently above the true level (positive bias). Hence statements (B) and (C) are correct.
Step 4: Instrument C.
Values are scattered, but shifted above the true level. It shows only positive bias but is not unbiased. Hence statement (D) is correct, (E) is incorrect.
Step 5: Conclusion.
Correct findings are B, C, and D only.

Step 1: Recall z-score formula.
where = observed value, = mean, = standard deviation.
Step 2: Substitute given values.
Here, , , .
Step 3: Interpretation.
A z-score of -1.5 means that a pregnancy lasting 36 weeks is 1.5 standard deviations below the mean gestation length.
Step 4: Conclusion.
Therefore, the z-score is -1.5.

Step 1: Recall the formula for Standard Error (SE).
The standard error of the mean is given by:
where is the sample standard deviation and is the number of individuals.
Step 2: Substitute the given values.
Here, and .
Step 3: Solve for .
Step 4: Conclusion.
Thus, to reduce the standard error of the mean to 1 mm Hg, at least 106 individuals are required.
Final Answer:

Step 1: Distinction between random and systematic errors.
- Random errors: Caused by chance fluctuations, reduced by larger sample size or statistical tests.
- Systematic errors (bias): Persistent, non-random errors due to flaws in design, data collection, or analysis.
Step 2: Systematic error types.
- Confounding bias: Occurs when a third variable distorts the association between exposure and outcome.
- Information bias: Caused by misclassification or errors in measurement of variables.
- Selection bias: Occurs when the method of selecting participants produces a sample not representative of the target population.
Step 3: Eliminating incorrect terms.
- Direction bias: Not a standard recognized form.
- Random bias: Contradictory term, since bias is systematic, not random.
Step 4: Conclusion.
Systematic errors include confounding bias, information bias, and selection bias.

Step 1: Apply Bayes’ theorem.
We are asked to calculate the Positive Predictive Value (PPV), i.e., .
Bayes’ theorem:
Step 2: Substitute values.
- Sensitivity
- False Positive Rate
- Prevalence
-
Step 3: Calculate numerator and denominator.
Numerator =
Denominator =
Step 4: Final probability.
Step 5: Conclusion.
The probability that a person truly has the disease given a positive test is 15.14%.

Step 1: Recognize distribution type.
This is a binomial probability problem because there are a fixed number of independent trials (4 patients), each with two outcomes (success or failure), and a constant probability of success ( ).
Step 2: Write binomial probability formula.
The probability of exactly successes in trials is:
Step 3: Substitute values.
Here, , , , and .
Step 4: Simplify calculation.
Step 5: Correct option.
The correct probability is 0.0486, which matches option (D).
% Correction Note Note: If the exam’s official key lists 0.324, that corresponds to the probability of 3 successes, not 2. Please double-check the key.