#StackBounty: #confidence-interval #proportion #error-propagation How do I calculate error margins for difference in proportions?

Bounty: 50

I have data that captures responses to stimuli and categorises both the cue stimuli and response stimuli for both male and female, as follows:

Cue Stimulus Response Stimulus Gender Cue Count Cue Proportion Response Count Response Proportion
A A F 5,365 6.74% 7,565 9.5%
A A M 3,588 4.32% 6,102 10.40%
B B F 3,443 6.74% 4,669 5.86%
B B M 2,598 4.43% 3,177 5.42%

This means that Females have been given cue Stimulus category A 5,365 times (6.74% of female cues). Females have responded with Stimulus category A 7,565 times (9.5% of female responses). The overall number of cue-response pairs is $79,625 F$ and $58,645 M$. The responses were given by $37,188$ female and $24,732$ male participants.

I want to understand if Females or Males have tendencies to respond more frequently in any of the categories.

I have attempted to calculate an error margin for each row based on another question:

$$
se = sqrt{
frac{pq}{n}
}
$$

Would the error margin for 1.96 Confidence Interval be as follows, for the first row of data?

$$
E = pmsqrt{0.095*(1-0.095) / 79,625}*1.96 = pm0.2%
$$

I then calculated the difference between the cue proportions and the response proportions. e.g. the first line difference would be $9.5% – 6.74% = 2.76%$. Is it valid to use the error margin obtained above for this result? e.g. $2.76% pm 0.2%$


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