Question

Over a period of 12 months, the new water purifier is tested to see if the quality if improved by testing the quality of water. The purity is measured on scale of 0-5 and the results are measured, and the matched differences are computed. The sample mean(difference) is 0.52 with a sample standard deviation of 0.34. Test the claim that the new purifier is no different to the old one at 10% level of significance.

a.Details given in problem

b.Assumptions if any

c.Null Hypothesis, Alt Hypothesis, and the tailed test

d.Critical Statistic

e.Compute test statistic

f.Compute p-value

g.At ___% level of significance, we have ______ evidence to reject Null Hypothesis

h.Since p-value ____ α at ___% level of significance, we have ______evidence to reject the Null Hypothesis

Answer 1

a. We are given the sample mean(difference) and standard deviation.

Here, we will be using Paired T -test to test the hypothesis.

b. The paired sample t-test has four main assumptions:

• The dependent variable must be continuous (interval/ratio).
• The observations are independent of one another.
• The dependent variable should be approximately normally distributed.
• The dependent variable should not contain any outliers.

c. Null Hypothesis : The new purifier is no different to the old one.

Alternaive Hypothesis : The new purifier is different to the old one.

This is a One-tailed test , since we want to test whether the purity has increased or not.

d. Critical statistic = t_n-1,

We have , n = 12 and = 0.10 ( level of significance)

t_12-1,0.10 = t_11,0.10 = 1.363

e. Test statistic = Mean difference / ( Standard deviation / n^0.5 ) = 0.52 / ( 0.34 / 12^0.5) = 5.298

f. P-value = P(t > 5.298) = 0.000127

g. At 10% level of significance, we have sufficient evidence to reject Null Hypothesis.

h. Since p-value is less than at 10% level of significance, we have sufficient evidence to reject the Null Hypothesis.