Question

The accompanying data on degree of spirituality for a random sample of natural scientists and a random sample of social scientists working at research universities appeared in a paper. Is there evidence that the spirituality category proportions are not the same for natural and social scientists? Test the relevant hypotheses using a significance level of α = 0.01.

	Degree of Spirituality
	Very	Moderate	Slightly	Not at All
Natural Scientists	56	160	195	215
Social Scientists	56	222	244	238

State the appropriate null and alternative hypotheses.

H₀: The spirituality category proportions are the same for natural scientists and social scientists.
H_a: The spirituality category proportions are not the same for natural scientists and social scientists.

H₀: The spirituality category proportions are not the same for natural scientists and social scientists.
H_a: The spirituality category proportions are the same for natural scientists and social scientists.   

H₀: Degree of spirituality and type of scientist are not independent.
H_a: Degree of spirituality and type of scientist are independent.

H₀: Degree of spirituality and type of scientist are independent.
H_a: Degree of spirituality and type of scientist are not independent.

Find the test statistic and P-value. (Use technology. Round your test statistic to three decimal places and your P-value to four decimal places.)

X²=

P-value=

State the conclusion in the problem context.

Reject H₀. There is convincing evidence to conclude that the spirituality category proportions are not the same for natural scientists and social scientists.

Reject H₀. There is not convincing evidence to conclude that the spirituality category proportions are not the same for natural scientists and social scientists.    

Fail to reject H₀. There is not convincing evidence to conclude that the spirituality category proportions are not the same for natural scientists and social scientists.

Fail to reject H₀. There is convincing evidence to conclude that the spirituality category proportions are not the same for natural scientists and social scientists.

Answer 1

We have test that is there any evidence that the spirituality category proportions are not the same for natural & social scientists.

Hypothesis:

Null hypothesis: H₀ - The spirituality category proportions are the same for natural scientists & social scientists.

Alternative hypothesis: H₁ - The spirituality category proportions are not the same for natural scientists & social scientists.

Test statistic -

Test criterion-

If p-value is less than 0.01 then reject H₀ at 1% level of significance.

Observation table -

		Degree of	Spirituality
	Very	Moderate	Slightly	Not at all	Total
Natural scientists	56	160	195	215	626
Social scientists	56	222	244	238	760
Total	112	382	439	453	1386

Calculation -

where, n_r = total of rth row, n_c = total of cth column, n = total sum

So,

Observation table-

(r,c)	E(r,c)	O(r,c)	d=(O-E)^2	d/E
(1,1)	50.5858	56	29.31356	0.579482
(1,2)	172.5339	160	157.0986	0.910538
(1,3)	198.2784	195	10.74791	0.054206
(1,4)	204.6017	215	108.1246	0.528464
(2,1)	64.4141	56	70.79708	1.099093
(2,2)	209.466	222	157.1012	0.750008
(2,3)	240.7215	244	10.74856	0.044651
(2,4)	248.3982	238	108.1226	0.435279
Total				4.401721

So,

P-value -

We use the chi square distribution calculator to find P(>4.4017)=0.22.

p = 0.22

Conclusion -

Since, p > 0.01 so we accept H₀ at 1% level of significance.

Failed to reject H₀, There is not convincing evidence to conclude that the spirituality category proportions are not the same for natural scientists & social scientists.