Question

For finding p-values, there are two approaches - bootstrap or use a distribution for the test statistic under the null hypothesis (like standard normal, T, Chi-Square, F). Using the distribution approach requires some assumptions (such as normality, or approximate normality, of the population distribution, large sample sizes, etc.) For the following testing scenarios, write down the assumptions that are needed:

a. Testing of equality of proportions between two categorical variables - p1 and p2. Ho: p1 = p2 vs H1: p1 /= p2. Sample size n1 and n2

b. Testing for equality of proportions between three categorical variables using the Chi-Square Goodness of Fit approach Ho: p1 = p2 = p3 vs H1: at least one pair of pi's is not the same. Sample sizes n1, n2, n3. Total sample size n = n1+n2+n3

a. Testing of equality of population means between three populations using the ANOVA table. Ho: mu1 = m2 = mu3 vs H1: not all population means are the same Sample size n1, n2, n3

Answer 1

Answer:-

(1)The test procedure, called the two-proportion z-test, is appropriate when the following conditions are met:

• The sampling method for each population is simple random sampling.
• The samples are independent.
• Each sample includes at least 10 successes and 10 failures.
• Each population is at least 20 times as big as its sample.

(2) The chi-square goodness of fit test is appropriate when the following conditions are met:

• The sampling method is simple random sampling.
• The variable under study is categorical.
• The expected value of the number of sample observations in each level of the variable is at least 5 .

(3)To use the ANOVA test we made the following assumptions:

• Each group sample is drawn from a normally distributed population
• All populations have a common variance
• All samples are drawn independently of each other
• Within each sample, the observations are sampled randomly and independently of each other
• Factor effects are additive