Applied Statistics

Numerical 1: Arithmetic Mean

Question: Calculate the arithmetic mean of the observations: 5, 8, 10, 12 and 15.

x = 5, 8, 10, 12, 15

Σx = 5 + 8 + 10 + 12 + 15 = 50

n = 5

Numerical 2: Median

Question: Find the median of: 4, 7, 2, 9, 6, 5, 8.

Data = 4, 7, 2, 9, 6, 5, 8

2, 4, 5, 6, 7, 8, 9

Numerical 3: Standard Deviation

Question: Calculate the standard deviation of: 2, 4, 6, 8 and 10.

x = 2, 4, 6, 8, 10

Numerical 4: Coefficient of Variation

Question: The mean of a dataset is 50 and its standard deviation is 5. Calculate the coefficient of variation.

Mean = 50
SD = 5

Numerical 5: Mean from Frequency Distribution

Question: Calculate the mean from the following frequency distribution.

Numerical 1: Pearson's Correlation

Question: Calculate Karl Pearson's correlation coefficient for:

n = 5
Σx = 15
Σy = 21
Σx² = 55
Σy² = 97
Σxy = 73

Numerical 2: Spearman's Rank Correlation

Question: Calculate Spearman's rank correlation coefficient when the ranks are:

Differences in ranks:

Σd² = 4

Numerical 3: Interpretation of Correlation

Question: A study reports a correlation coefficient of r = −0.82 between two biological variables. Interpret the result.

The negative sign indicates an inverse relationship between the two variables. The magnitude, 0.82, indicates a strong linear association.

Numerical 1: Simple Linear Regression

Question: For the regression equation

predict Y when X = 10.

a = 5
b = 2
X = 10

Numerical 2: Residual

Question: If the observed value is 28 and the predicted value is 25, calculate the residual.

Numerical 3: Coefficient of Determination

Question: If the correlation coefficient between X and Y is 0.8, calculate R² and interpret it.

Numerical 1: Expected Frequency

Question: In a contingency table, the row total is 40, the column total is 30, and the grand total is 100. Calculate the expected frequency.

Numerical 2: Chi-square Statistic

Question: If observed frequency is 18 and expected frequency is 15, calculate the contribution of the cell to χ².

Numerical 3: Yule's Coefficient

Question: For a 2 × 2 table, let a = 20, b = 10, c = 5 and d = 15. Calculate Yule's coefficient of association.

Numerical 1: Simple Probability

Question: A box contains 10 bacterial culture tubes, of which 4 are positive. If one tube is selected randomly, find the probability that it is positive.

Numerical 2: Addition Theorem

Question: If P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2, calculate P(A ∪ B).

Numerical 3: Conditional Probability

Question: If P(A ∩ B) = 0.20 and P(B) = 0.50, find P(A|B).

Numerical 4: Bayes' Theorem

Question: A disease occurs in 10% of a population. A test is positive in 90% of diseased individuals and in 5% of non-diseased individuals. Find the probability that a person actually has the disease given that the test is positive.

P(D) = 0.10
P(+|D) = 0.90
P(+|D̄) = 0.05
P(D̄) = 0.90

Numerical 1: Binomial Distribution

Question: If the probability of success is 0.4, calculate the probability of exactly 2 successes in 5 independent trials.

n = 5
x = 2
p = 0.4
q = 0.6

Numerical 2: Binomial Mean and Variance

Question: For a binomial distribution with n = 20 and p = 0.3, calculate the mean and variance.

Numerical 3: Poisson Distribution

Question: If the average number of bacterial colonies observed in a given sample is 3, calculate the probability of observing exactly 2 colonies under a Poisson model.

λ = 3
x = 2

Numerical 4: Normal Distribution

Question: A variable has mean 50 and standard deviation 10. Find the Z-score corresponding to X = 70.

Numerical 1: Confidence Interval for Mean

Question: A sample has mean 50, population standard deviation 10 and sample size 100. Calculate the 95% confidence interval for the population mean.

x̄ = 50
σ = 10
n = 100
Z = 1.96

Numerical 2: Confidence Interval for Proportion

Question: In a sample of 100 individuals, 60 are positive for a particular test. Calculate the approximate 95% confidence interval for the population proportion.

p = 60/100 = 0.60
q = 0.40
n = 100
Z = 1.96

Numerical 1: One-Sample Z Test

Question: A population has hypothesized mean 50 and known standard deviation 10. A sample of 100 has mean 52. Test the null hypothesis at the 5% significance level.

μ = 50
x̄ = 52
σ = 10
n = 100
α = 0.05

H₀: μ = 50
H₁: μ ≠ 50

At 5% significance level for a two-tailed test, the critical value is approximately ±1.96.

Numerical 2: One-Sample t Test

Question: A sample of 10 observations has mean 24 and standard deviation 4. Test whether the population mean differs from 20 at the 5% level.

x̄ = 24
μ = 20
s = 4
n = 10

H₀: μ = 20
H₁: μ ≠ 20

Degrees of freedom:

Numerical 3: F Test

Question: Two samples have variances 25 and 10. Calculate the F statistic by placing the larger variance in the numerator.

Numerical 4: One-Way ANOVA

Question: Three groups have the following observations:

The purpose of ANOVA is to determine whether the group means differ significantly.

Group A mean = 9
Group B mean = 12
Group C mean = 15

The complete ANOVA calculation requires the between-group and within-group sums of squares.

Numerical 1: Sign Test

Question: In a paired study, 8 observations show a positive difference and 2 show a negative difference. Ignoring ties, determine the number of positive and negative signs.

Positive differences = 8
Negative differences = 2
Total = 10

Numerical 2: Wilcoxon Signed-Rank Test

Question: Consider paired differences:

+2, −1, +4, +3, −2

There are no zero differences.

Positive rank sum:

Negative rank sum:

Numerical 3: Mann–Whitney U Test

Question: Two independent groups have the following observations:

Group A: 4, 7, 8
Group B: 2, 5, 6

R₁ for Group A = 2 + 5 + 6 = 13

R₂ for Group B = 1 + 3 + 4 = 8

Numerical 4: Kruskal–Wallis Test

Question: Three independent groups contain the following observations:

The combined observations are: 2, 3, 4, 5, 6, 7, 8, 9, 10.

Their ranks are therefore 1 through 9.

Group A:

Group B:

Group C:

Here N = 9 and each group contains 3 observations.