Question

1. 50 men and 50 women are intercepted at a mall, asked to smell two fragrances (vanilla vs. musk) and then pick the one they like best. The data are shown below. Is gender related to preference for a certain type of fragrance? Use alpha = .05.

	Musk	Vanilla
Men	35	15
Women	20	30

Step 1: State the null and alternative hypotheses

Step 2: Indicate the critical value of the appropriate statistic (and draw a picture of the distribution, showing the alpha/critical region and the critical value of the statistic)

Step 3: Calculate the test statistic/p-value.

Step 4: Decision and conclusion (do you retain or reject the null hypothesis? And what does that mean in terms of the problem - write a verbal conclusion that relates back to the actual problem.

Answer 1

Data Summary

Observed Frequencies (O)	Musk	Vanilla	Total
Men	35	15	50
Women	20	30	50
Total	55	45	100

Step 1 :

The null and alternative hypotheses are          
Ho :   Gender is not related to preference of fragrance          
Ha :   Gender is related to preference of fragrance         

Step 2:

From chi square table or CHISQ.INV.RT(alpha, df) function of Excel      
we find the critical value of chi square      
Critical χ² = CHISQ.INV.RT(0.05, 1) = 3.8415      
Critical χ² = 3.8415      

Step 3 :

We use Chi Square test of independence          
          
Grand Total of frequencies = 100          
To find Expected Frequencies          
Expected Frequencies = (Row Total * Column Total)/Grand Total          

Expected Frequencies (E)	Musk	Vanilla
Men	27.5	22.5
Women	27.5	22.5

Following table gives the value of (Observed - Expected)² / Expected

	Musk	Vanilla
Men	2.0455	2.5
Women	2.0455	2.5

Chi Square Value = ∑[(Observed - Expected)² / Expected]          
χ² test statistic = 9.0909          
          
Degrees of freedom = df = (number of rows - 1) * (number of columns - 1)          
Degrees of freedom = df = 1          
          
From chi square table or CHISQ.DIST.RT(X, df) function of Excel          
we find the p-value          
p-value = CHISQ.DIST.RT( 9.0909, 1) = 0.0026          
p-value = 0.0026        

Step 4:

Decision :  
0.0026 < 0.05  
that is p-value <= α  
Hence we REJECT Ho  
  
9.0909 > 03.8415  
that is test statistic χ² > Critical χ²  
Hence we REJECT Ho  
  
Conclusion :
There exists enough statistical evidence at α = 0.05 to show that gender is related to preference of fragrance