statistics hypothesis testing

Question

 Pain medications sometimes come with side effects. One side effect is dizziness. A researcher wanted to determine if some pain medications produced more or less dizziness than others. The results from his study are shown in the table below. Use a test to determine if the proportion of people within each treatment group who experienced dizziness are the same at the αα = .01 level of significance.

	Tordol	Placebo	Naproxen	Tylenol	Advil
Dizziness	83	32	36	5	8
No dizziness	4063	1832	1330	382	337

1) Determine the null and alternative hypotheses. Select the correct pair.

1. Ho:Ho: The proportion of people within each treatment group who experienced dizziness are the same 
2. Ha:Ha: The proportion of people within each treatment group who experienced dizziness are different
3. Ho:Ho:The proportion of people within each treatment group who experienced dizziness are independent 
4. Ha:Ha: The proportion of people within each treatment group who experienced dizziness are dependent

2) Determine the test Statistic. Round your answer to two decimals.

χ2χ2 = 

3) Determine the p-value. Round your answer to four decimals.

p-value = 

4) Make a decision.

1. Fail to reject the null hypothesis
2. Reject the null hypothesis

5) Pick a conclusion.

1. There is sufficient evidence to support the claim that proportion of people within each treatment group who experienced dizziness are different
2. There is not sufficient evidence to support the claim that the proportion of people within each treatment group who experienced dizziness are different

Answer 1

A chi-square test for homogeneity is needed to determine whether the proportion of people experiencing dizziness is the same across different pain medications at a 1% level of significance.

1.) (Null Hypothesis) H₀: The proportion of dizziness experiences is the same for all pain medications.

(Alt. Hypothesis) H₁: At least one medication has a different proportion of dizziness experiences.

2.) You will need to create the table to find the expected frequencies for each cell.

Expected Frequency (E) = (Row Total*Column Total) / Grand Total

E(Dizziness, Tordal) = (164*4146) / 8108 = 83.86

E(Dizziness, Placebo) = (164*1864) / 8108 = 37.7

E(Dizziness, Naproxen) = (164*1366) / 8108 = 27.63

E(Dizziness, Tylenol) = (164*387) / 8108 = 7.83

E(Dizziness, Advil) = (164*345) / 8108 = 6.98

E(No Dizziness, Tordal) = (7944*4146) / 8108 = 4,062.14

E(No Dizziness, Placebo) = (7944*1864) / 8108 = 1,826.3

E(No Dizziness, Naproxen) = (7944*1366) / 8108 = 1,338.37

E(No Dizziness, Tylenol) = (7944*387) / 8108 = 379.17

E(No Dizziness, Advil) = (7944*345) / 8108 = 338.022

O(Dizziness, Tordal) = 83

O(Dizziness, Placebo) = 32

O(Dizziness, Naproxen) = 36

O(Dizziness, Tylenol) = 5

O(Dizziness, Advil) = 8

O(No Dizziness, Tordal) = 4,063

O(No Dizziness, Placebo) = 1,832

O(No Dizziness, Naproxen) = 1,330

O(No Dizziness, Tylenol) = 382

O(No Dizziness, Advil) = 337

Find the difference, then square each one:

O - E(Dizziness, Tordal) = 83 - 83.86 = -0.86 ⇒ 0.7396

O - E(Dizziness, Placebo) = 32 - 37.7 = -5.7 ⇒ 32.49

O - E(Dizziness, Naproxen) = 36 - 27.63 = 8.37 ⇒ 70.0569

O - E(Dizziness, Tylenol) = 5 - 7.83 = -2.83 ⇒ 8.0089

O - E(Dizziness, Advil) = 8 - 6.98 = 1.02 ⇒ 1.0404

O - E(No Dizziness, Tordal) = 4,063 - 4,062.14 = 0.86 ⇒ 0.7396

O - E(No Dizziness, Placebo) = 1,832 - 1,826.3 = 5.7 ⇒ 32.49

O - E(No Dizziness, Naproxen) = 1,330 - 1,338.37 = -8.37 ⇒ 70.0569

O - E(No Dizziness, Tylenol) = 382 - 379.17 = 2.83 ⇒ 8.0089

O - E(No Dizziness, Advil) = 337 - 338.02 = -1.02 ⇒ 1.0404

Divide the square difference by the expected value for each cell:

(O - E)² / E (Dizziness, Tordal) = 0.7396 / 83.86 = 0.008819

(O - E)² / E (Dizziness, Placebo) = 32.49 / 37.7 = 0.8618

(O - E)² / E (Dizziness, Naproxen) = 70.0569 / 27.63 = 2.5355

(O - E)² / E (Dizziness, Tylenol) = 8.0089 / 7.83 = 1.02285

(O - E)² / E (Dizziness, Advil) = 1.0404 / 6.98 = 0.1491

(O - E)² / E (No Dizziness, Tordal) = 0.7396 / 4062.14 = 0.0001821

(O - E)² / E (No Dizziness, Placebo) = 32.49 / 1826.3 = 0.01779

(O - E)² / E (No Dizziness, Naproxen) = 70.0569 / 1338.37 = 0.05234

(O - E)² / E (No Dizziness, Tylenol) = 8.0089 / 379.17 = 0.02112

(O - E)² / E (No Dizziness, Advil) = 1.0404 / 338.02 = 0.003078

Add them up and you will get χ² = 4.6725791.

The critical value from the table is degrees of freedom = (number of rows - 1) * (number of columns - 1). We have df = (2-1)*(5-1) = 1*4 = 4 and its p-value is 0.322559. Since the p-value is greater than 0.01, we fail to reject the null hypothesis. Therefore, the proportion of dizziness experiences is the same for all pain medications.