How to calculate the statistical significance of the difference in a categorical value of two populations?

There has recently been a statistical report regarding whether Cypriots are ready for a female president. There was a difference in the opinions of men and women and I'm wondering if it is statistically significant.

The sample size is 800 and based on the numbers it seems that 50% are men and 50% are women (this was not reported but calculated based on the numbers provided).

Out of 800 people 60% said yes, 38% said no, and 2% N/A

63% of men said yes, 35% said no and 2% N/A

57% of women said yes, 41% said no and 2% N/A

How should I go about doing the analysis? I'm not a statistician and I never really used this stuff but it seems to me that I should do a chi-squared test. However, I'm not sure what the expected values should be here.

So how can I determine if there is actually a statistically significant difference between the opinion of men and women?

• The "expected values" to use for the chisquared test (really the estimated expected values), are the ones you can calculate, assuming the nul hypothesis is true, thatis, that there is no difference! May 11, 2022 at 14:44
• @kjetilbhalvorsen thank you for your reply. So in this case it should be compared to what men said and not the average of the entire population correct? That is actually what I did but I did not feel very confident with my decision. The reason I was not confident is because it seems to me that this assumes that the sample of men is large enough. May 11, 2022 at 18:23
• btw, I think the way the question was asked is a bit misleading May 11, 2022 at 18:25
• Misleading in what respect? If you have questions not already answered here, now would be the time to ask. May 12, 2022 at 15:27
• @BruceET no I meant the way the question was phrased by whoever made the study was misleading. At the first glimpse it seems that women are the ones who say they are not ready for a female president. However, after taking another look they are actually saying that Cypriots (as a society) are not ready. At least that is how I saw it. May 13, 2022 at 16:25

One method is to do a chi-squared test. However, it requires count--not percentages. So, I'll assume, as you did, that 400 of the 800 are men and 400 are women.

This leads to the following $$2\times 3$$ table of counts, where columns are for Yes, No, N/A, respectively.

men = c(248, 140, 8)
wom = c(228, 164, 8)
TAB = rbind(men,wom)
TAB
[,1] [,2] [,3]
men  248  140    8
wom  228  164    8

Pearson's chi-squared test of homogeneity (declining Yates' Correction on account of reasonably large sample size) gives P-value $0.2573 > 0.05 = 5% and so no significant differences are found between opinions of men and women. chisq.test(TAB) Pearson's Chi-squared test data: TAB X-squared = 2.715, df = 2, p-value = 0.2573 You asked about the expected counts for the test. Based on the null hypothesis of homogeneous opinions, the expected count in each cell of the table is found by multiplying its row total by its column total and the dividing by the table total $$796;$$ do not round expected counts to integers. [The total is 796 instead of 800, because the percentages you gave for each row add to 99%, and I assumed a few people failed to participate. I will leave this for you to resolve.] The following R code prints out the expected counts used in the chi-squared test. chisq.test(TAB)$exp
[,1]     [,2]     [,3]
men 236.804 151.2362 7.959799
wom 239.196 152.7638 8.040201

I hope you will follow through the formulas in your text, do the computations yourself, and compare the output of the test from R, with results from your own computations.