Question

Do men and women select different breakfasts? The breakfasts ordered by randomly selected men and women at a popular breakfast place are shown below.

Frequencies of Breakfasts for Men and Women

	French Toast	Pancakes	Waffles	Omelets
Men	93	51	86	71
Women	66	67	100	50

What can be concluded at the α = 0.01 significance level?

1. What is the correct statistical test to use?
   • Goodness-of-Fit
   • Paired t-test
   • Homogeneity
   • Independence
2. What are the null and alternative hypotheses?
   H0:
   • Breakfast ordered and gender are independent.
   • The distribution of breakfast ordered for men is the same as it is for women.
   • The distribution of breakfast ordered for men is not the same as it is for women.
   • Breakfast ordered and gender are dependent.
   
   H1:
   • Breakfast ordered and gender are independent.
   • The distribution of breakfast ordered for men is the same as it is for women.
   • The distribution of breakfast ordered for men is not the same as it is for women.
   • Breakfast ordered and gender are dependent.
3. The test-statistic for this data = ______ (Please show your answer to three decimal places.)  
4. The p-value for this sample = _____ (Please show your answer to four decimal places.)  
5. The p-value is? (greater than, less than or equal to) α  
6. Based on this, we should
   • reject the null
   • accept the null
   • fail to reject the null
7. Thus, the final conclusion is...
   • There is sufficient evidence to conclude that breakfast ordered and gender are dependent.
   • There is insufficient evidence to conclude that the distribution of breakfast ordered for men is not the same as it is for women.
   • There is sufficient evidence to conclude that the distribution of breakfast ordered for men is not the same as it is for women
   • There is insufficient evidence to conclude that breakfast ordered and gender are dependent.
   • There is sufficient evidence to conclude that the distribution of breakfast ordered for men is the same as it is for women.

Answer 1

The correct statistical test to use is,

Chi square test for independence.

The null and alternative hypothesis are,

H₀: Breakfast ordered and gender are independent.

H₁: Breakfast ordered and gender are dependent.

The test statistic is,

x? (O; - E;) Ei

Where, O_i= Observed frequency

E_i= Expected frequency

The observed frequencies are,

	French Toast	Pancakes	Waffles	Omelets	Total
Male	93	51	86	71	301
Female	66	67	100	50	283
Total	159	118	186	121	584

The expected frequencies are,

	French Toast	Pancakes	Waffles	Omelets	Total
Male	81.95	60.82	95.87	62.36	301
Female	77.05	57.18	90.13	58.64	283
Total	159	118	186	121	584

Observed(Oi)	Expected(Ei)	x? (O; - E;) Ei
93	81.95	1.49
51	60.82	1.59
86	95.87	1.02
71	62.36	1.19
66	77.05	1.59
67	57.18	1.69
100	90.13	1.08
50	58.64	1.27
		$\chi ^2_{cal}=10.92$

$\therefore$ Test statistic for this data= 10.920

Degrees of freedom= (row-1)*(coloumn-1)=(2-1)*(4-1)=3

From $\chi ^2$ table with 3 df and ,

α = 0,01

p-value for this sample= 0.0122

p-value is greater than $\alpha$ . That is 0.0122>0.01

We reject H₀ if p-value< $\alpha$

Based on this, we should fail to reject the null hypothesis.

Thus the final conclusion is,

There is insufficient evidence to conclude that breakfast ordered and gender are dependent.

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