a) In a sample of 1000 people in Maharashtra, 540 are rice eaters and the rest are wheat eaters.
Can we assume that both rice and wheat eaters are equally popular in this state at 1% level of
significant?
To determine if we can assume that both rice and wheat eaters are equally popular in Maharashtra, we can use a chi-squared test of independence.
Null hypothesis (H0): The proportions of rice and wheat eaters are equal in Maharashtra.
Alternative hypothesis (HA): The proportions of rice and wheat eaters are not equal in Maharashtra.
We have the following information:
- Sample size: 1000 people
- Rice eaters: 540 people
- Wheat eaters: (1000-540) = 460 people
Using these numbers, we can calculate the expected number of rice and wheat eaters in a sample of 1000 people under the assumption of equal proportions.
Expected number of rice eaters: (1000/2) = 500
Expected number of wheat eaters: (1000/2) = 500
Now, we can calculate the chi-square statistic:
chi-square = ( (observed rice - expected rice)^2 / expected rice ) + ( (observed wheat - expected wheat)^2 / expected wheat )
= ( (540 - 500)^2 / 500 ) + ( (460 - 500)^2 / 500 )
= ( (40)^2 / 500 ) + ( (-40)^2 / 500 )
= 1600/500 + 1600/500
= 3.2 + 3.2
= 6.4
Using a chi-square distribution lookup table or a statistical software, we would find the critical chi-square value for a 1% level of significance with 1 degree of freedom. For a one-sided test, the critical value is approximately 6.635.
As our calculated chi-square value (6.4) is below the critical value (6.635), we fail to reject the null hypothesis. Therefore, we cannot assume that both rice and wheat eaters are equally popular in Maharashtra at a 1% level of significance.