Question

The accompanying data on degree of spirituality for a sample of natural scientists and a sample of social scientists working at research universities appeared in a paper. Assume that it is reasonable to regard these two samples as representative of natural and social scientists at research universities. 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 0.01. Degree of Spirituality Very Moderate Slightly Not at All Natural Scientists 54 161 195 216 Social Scientists 55 220 239 242 x² = _________ P-value= ____________

Answer 1

There is no evidence that the spirituality category proportions are different for natural and social scientists.

Explanation:

To solve this question we must perform a Chi square test calculating the expected values of the observed behavior.

We start by completing the table given by adding the totals:

Observed Very Moderate Slightly Not at all Total

Natural Sc 54 161 195 216 626

Social Sc 55 220 239 242 756

Total 109 381 434 458 1382

Now, the chi square value is calculated with the following formula:

`x^(2)`=∑∑
`(O_(ij)-E_(ij) )/(E_(ij) )`

Where:

`O_(ij)`: Observed value (the ones we have in our table)

`E_(ij)`: Expected value

The expected value of every observation (ij) is calculated as it follows:

`E_(ij)=(n_(i)c_(j) )/(N)`

Where,

`n_(i)`: marginal total by rows

`c_(j)`: marginal total by columns

N: Total of observations

Now, for the expected observations we obtain the following table:

Expected Very Moderate Slightly Not at all Total

Natural Sc 49.37 172.58 196.59 207.46 626

Social Sc 59.63 208.42 237.41 250.54 756

Total 109 381 434 458 1382

Having the expected values we now can calculate
`x^(2)` by first calculating
`(O_(ij)-E_(ij) )/(E_(ij) )` for each observation:

Chi sq Very Moderate Slightly Not at all Total

Natural Sc 0.434 0.777 0.013 0.352 1.575

Social Sc 0.359 0.643 0.011 0.291 1.304

Total 0.793 1.420 0.024 0.643 2.879

`x^(2)=2.879`

We may consider our null hypothesis as it follows:

`H_(0):` The degree of spirituality category proportions are the same for natural and social scientists.

To prove this we have:

`P(x^(2)\geq 2.879)`

α=0.01

α: significance level

Calculating this value with a chi square table (or with statistical software like R) we obtain:

P-value=0.4105

Because the p-value is larger than α the null hypothesis is accepted. Which means we cannot say that the spirituality category proportions are different.