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Suppose I am an inactive person - I go for a walk on average a few times a .month (e.g. every day there is a 5% chance I go for a walk, independent of yesterday) . However, each time I go for a walk, I meet this man with a red hat.

  1. Can statistics be used to infer that this man walks more often than I do? Or perhaps his walks are also decided by a random process? Knowing that I go on such few walks but still see this man each time, it seens more likely that he walks more often than me?

  2. Suppose I go for walks every Sunday and still see this man. Is it more logical for me to believe that this man might have a busy schedule (a schedule similar to mine, e.g. we both work 6 days a week) and simply prefers to only walk on Sundays? (I.e. the other days of the week he doesn't walk)

Thanks

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Statistics can absolutely be utilized for these scenarios

1)
Assuming independence of walking schedules (Man is not more likely to be walking if you are).

Probability you go for a walk on any day = $P_{you}$
Probability man in red hat goes for a walk on any day = $P_{red}$

Your hypotheses would be:
$H_{0}: P_{red} = P_{you}$
$H_A: P_{red} > P_{you}$

There are a few tests you could perform here, but logically, the likelihood of the man in the red hat walking on the same day as you each time when his probability of going for a walk is also 0.05 is very small.

This can be demonstrated using the binomial distribution:
(All calculated under null hypothesis...)

  • $p_{\text{both walking}} = 0.05 * 0.05 = 0.025$
  • $p_{\text{at least 1 person isn't walking}} = 0.975$
  • Number of days you go for a walk = n
  • Number of days you see man in red hat while walking = x

$f(n,x) = ({n \atop x})0.025^{x}0.975^{n-x}$

As you can see, as the amount of days you walk increases (assuming you see man in red hat each day), the likelihood of this happening is extremely small. So we can infer the man's probability of walking any given day is higher than 0.05

2)

In this scenario, I'm unclear on what exactly you're looking to prove, given you only observe Sundays. If you could be more specific in the theoretical behavior of the man in the red hat, we could probably use similar tests for your assumptions.

  • Are we assuming the man in the red hat still has some probability of walking any day of the week?

  • Are we assuming the man only walks once a week, and trying to figure out if he walks the same day each week or a random day?

Hope this helps

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    $\begingroup$ It's critical to lay out the assumptions underlying any statistical test, which in the first case is independence of the two people's walking schedules. If the man in the red hat is a stalker who follows you everywhere, the fact that you see him every time you leave the house does not indicate that he walks any more than you do; it instead indicates that his walking schedule is not independent of yours. $\endgroup$ Sep 21 at 15:52
  • $\begingroup$ @NuclearHoagie you're absolutely right, I inferred independence from their question, I'll add that assumption to my answer $\endgroup$
    – jros
    Sep 21 at 15:54

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