As another example, NBA basketball players who are normally reliable free throw shooters are often saddled with a reputation of "choking" when missing a pair of free throws, especially when the stakes are high.   But is it necessarily choking?

Imagine a player who makes 86% of free throws.

If we assume (and this is a highly suspect assumption) that free throws are made independently as a Bernoulli process.. then the probability of missing two free throws in a row is Answer (give a decimal between 0 and 1 accurate to 4 decimal places).   Because this number represents a p-value, and is low,   it may be tempting to conclude that it is scientifically valid to conclude the player must have choked, by reject the null hypothesis of "no choking".

Although we cannot exclude choking as an explanation, it is misleading to use the p-value this way. Similar to the DNA example , when basketball players make many free throw attempts, mathematics can easily show likelihood of missing two in a row, eventually, becomes Answer.

If one "waits" for this occurrence and then pretends it was randomly sampled data, then one will be able to conclude that not only all basketball players choke, but also all robot free throw shooters and all random number generators choke.

It is never valid to construct the hypotheses from sampled data and then test that very same data against those hypotheses.     All data contain spurious patterns that are neither reproducible nor extendible as patterns to the population. It is misleading to find such a pattern and to then test the same data set for that pattern, for this would always lead to a significant result.

One must always test "future" data,   or data that was found independently of the information used construct the hypotheses.       Even when this is done (as it should be),   it is still possible to get a false positive result, but at least now we can safely assume the rate of false positive is controilled by the significance level ?α.

Question

As another example, NBA basketball players who are normally reliable free throw shooters are often saddled with a reputation of "choking" when missing a pair of free throws, especially when the stakes are high. But is it necessarily choking?

Imagine a player who makes 86% of free throws.

If we assume (and this is a highly suspect assumption) that free throws are made independently as a Bernoulli process.. then the probability of missing two free throws in a row is Answer (give a decimal between 0 and 1 accurate to 4 decimal places). Because this number represents a p-value, and is low, it may be tempting to conclude that it is scientifically valid to conclude the player must have choked, by reject the null hypothesis of "no choking".

Although we cannot exclude choking as an explanation, it is misleading to use the p-value this way. Similar to the DNA example , when basketball players make many free throw attempts, mathematics can easily show likelihood of missing two in a row, eventually, becomes Answer.

If one "waits" for this occurrence and then pretends it was randomly sampled data, then one will be able to conclude that not only all basketball players choke, but also all robot free throw shooters and all random number generators choke.

It is never valid to construct the hypotheses from sampled data and then test that very same data against those hypotheses. All data contain spurious patterns that are neither reproducible nor extendible as patterns to the population. It is misleading to find such a pattern and to then test the same data set for that pattern, for this would always lead to a significant result.

One must always test "future" data, or data that was found independently of the information used construct the hypotheses. Even when this is done (as it should be), it is still possible to get a false positive result, but at least now we can safely assume the rate of false positive is controilled by the significance level ?α.

Accepted Answer

Given:
Xi=p=0.86                   x=11-p=1-0.86        x=0
where Xi making free throw.