Hotter hand The basketball player in Exercise 55 has new...

Hotter hand The basketball player in Exercise 55 has new sneakers, which he thinks improve his game. Over his past 60 shots, he's made 32 - much better than the $50 \%$ he usually shoots. Do you think his chances of making a shot really increased? In other words, is making at least 32 of 60 shots really unusual for him? (Do you think it's his sneakers?)

Key Concepts

Binomial Distribution

This concept refers to the probability distribution that summarizes the likelihood that a value will take one of two independent outcomes under a given number of experiments. In scenarios where there are repeated independent trials with the same probability of success, such as making a shot in basketball, the binomial distribution provides a framework to calculate probabilities for different numbers of successes over a fixed number of trials.

Sampling Variability

Sampling variability is the natural variation in outcomes that occurs by chance when sampling from a population or conducting repeated experiments. Even if the underlying probability of success remains constant, you can observe fluctuations in outcomes from one set of trials to another, which means an observed result might not necessarily indicate a true change in ability or a difference in the underlying success rate.

Null Hypothesis Testing

This is a method used to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. In the context of performance improvement, the null hypothesis typically assumes that there is no change – for example, that the player's shooting percentage remains at its usual level – and testing is used to decide if the observed performance is significantly different from what is expected under this assumption.

P-Value

The p-value is a metric used in hypothesis testing that quantifies the probability of observing data at least as extreme as the actual observed data, under the assumption that the null hypothesis is true. A low p-value indicates that such an extreme result would be unlikely to occur by chance, suggesting that the null hypothesis may be rejected in favor of an alternative explanation.

Statistical Significance

Statistical significance is a determination of whether the observed effect in the data is unlikely to have occurred purely by chance. If a study's results are statistically significant, it suggests that the observed deviation from expected outcomes (using a standard threshold for p-values) is robust enough to imply a real phenomenon, rather than being due to random fluctuation in the data.

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