Latest questions in this category.
p-value 0.0023 is less than 0.01 but greater than 0.001, so result surpasses 1% significance threshold but not 0.1% level. By convention, p
Ref: Armitage P., Berry G. and Matthews J.N.S., Statistical Methods in Medical Research, 4th Edition, Chapter 4 Significance Tests and Confidence Intervals, interpretation of p-values thresholds 0.05 0.01 0.001 significance levels decision making, published by Blackwell Science.
Analysis of Variance partitions total variance into between-group and within-group components to test equality of means across multiple groups. While t-test handles two groups, ANOVA extends to more than two independent or related samples, controlling Type I error with F-ratio significance.
Ref: Fisher R.A., Statistical Methods for Research Workers, 14th Edition and Montgomery D.C., Design and Analysis of Experiments, 9th Edition Chapter 3 Analysis of Variance for testing difference among more than two means using F test, published by Wiley.
Open-end classes lack defined lower or upper limits, making range and standard deviation incalculable because mean depends on all values. Quartile deviation uses median and quartiles, which remain unaffected by extremes, thus provides robust dispersion measure for open-ended frequency distributions.
Ref: Gupta S.C. and Kapoor V.K., Fundamentals of Mathematical Statistics, 11th Edition, Chapter 2 Measures of Dispersion, quartile deviation as appropriate dispersion for open-end class distributions unaffected by extremes, published by Sultan Chand and Sons.
Total tablets = 4+3+5 =12. Underweight =4, so not underweight =8. Probability = favorable/total =8/12 =0.666... ≈0.67. Complement rule P(not A)=1-P(A)=1-4/12 confirms chance of drawing normal or overweight tablet together exceeds underweight probability.
Ref: Gupta S.C. and Kapoor V.K., Fundamentals of Mathematical Statistics, 11th Edition, Chapter 5 Probability Theory, classical probability favorable over total and complement rule for not underweight tablets calculation example, published by Sultan Chand and Sons.
Chi-square test of independence examines whether two categorical attributes are associated in contingency tables. It compares observed frequencies with expected under independence, following chi-square distribution. t, F, Z tests compare means or variances, not attribute independence for qualitative classifications.
Ref: Zar J.H., Biostatistical Analysis, 5th Edition, Chapter 23 Contingency Tables, chi-square test of independence for categorical attributes observed versus expected frequencies under null hypothesis, published by Pearson Prentice Hall.
Yule's Q or coefficient of association evaluates relationship between two binary attributes in a 2×2 contingency table. Unlike Pearson or Spearman correlations which need continuous or ordinal data, Yule's measure is specifically designed for qualitative attributes at two levels each.
Ref: Yule G.U. and Kendall M.G., An Introduction to the Theory of Statistics, 14th Edition, Chapter 3 Association of Attributes, Yule's coefficient of association Q for two binary attributes two by two contingency table, published by Charles Griffin and Company.
Eight percent defective implies 92 percent non-defective. Expected count = total × probability = 175 × 0.92 = 161. Expectation follows binomial mean n×p for non-defective proportion. Thus 14 defective tablets expected, leaving majority effective and meeting quality standards.
Ref: Montgomery D.C. and Runger G.C., Applied Statistics and Probability for Engineers, 6th Edition, Chapter 2 Probability Expectation, binomial expectation of non-defective proportion total times probability quality control defective tablets, published by John Wiley and Sons.
When mean exceeds median which exceeds mode, tail extends to right, indicating positive skew. Here 135 > 133 > 130 satisfies Mean > Median > Mode, characteristic of positively skewed distribution. Symmetric case would show equality, negative skew reverses order.
Ref: Snedecor G.W. and Cochran W.G., Statistical Methods, 8th Edition, Chapter 3 Frequency Distributions and Measures of Central Tendency, relation mean median mode indicates positive skew when mean greater than median greater than mode, published by Iowa State University Press.
Karl Pearson's coefficient of skewness = (Mean – Mode)/SD measures asymmetry. Substituting mean 10, mode 13, SD 1.5 yields (10-13)/1.5 = -3/1.5 = -2. Negative value indicates left skew, consistent with mean less than mode in this dataset.
Ref: Gupta S.C. and Kapoor V.K., Fundamentals of Mathematical Statistics, 11th Edition, Chapter 10 Skewness, Karl Pearson coefficient of skewness formula mean minus mode divided by standard deviation interpretation sign, published by Sultan Chand and Sons.
Empirical relationship Mode = 3 Median – 2 Mean approximates moderately skewed distributions. Substituting median 22 and mean 24 gives Mode = 66 – 48 = 18. This Pearson formula estimates central tendency linkage when distribution deviates slightly from symmetry.
Ref: Kenney J.F. and Keeping E.S., Mathematics of Statistics Part One, 3rd Edition, Chapter 4 Empirical Relation among Mean Median Mode, Pearson formula Mode equals three median minus two mean for moderately skewed distribution, published by Van Nostrand.
Two independent random samples from different hospitals require comparison of means using unpaired or independent samples t-test. Paired test needs matched pairs, Z-test needs large known variance. Here hospital groups are unrelated, so two-sample t-test for unequal samples is appropriate.
Ref: Rosner B., Fundamentals of Biostatistics, 8th Edition, Chapter 8 Hypothesis Testing Two Sample Inference, unpaired t-test for comparing means of two independent hospital samples with unknown variances, published by Cengage Learning.
Negatively skewed distributions have a long left tail pulling mean toward low values. Consequently mean becomes smallest, mode largest as peak shifts right, and median lies between. The classic inequality Mean < Median < Mode characterizes left-skewed data, opposite of positively skewed pattern.
Ref: Gupta S.P., Statistical Methods, 46th Edition, Chapter 6 Skewness Kurtosis and Moments, interpretation of negatively skewed distribution mean less than median less than mode left tail characteristics, published by Sultan Chand and Sons.
