Question

A basketball player who ordinarily makes about \(55 \%\) of his free throw shots has made 4 in a row. Is this evidence that he has a "hot hand" tonight? That is, is this streak so unusual that it means the probability he makes a shot must have changed? Explain.

Short answer

Not necessarily; 4 consecutive shots is not rare enough to suggest a 'hot hand.'

Step-by-step solution

Understanding the Problem

A basketball player has a historical success rate of 55% for free throw shots. He has successfully made 4 consecutive shots. We need to determine if this streak is statistically unusual enough to suggest his probability of making a shot has changed, indicating a 'hot hand'.

Define Assumptions and Hypotheses

Under the assumption that his ability has not changed, the probability of making a shot remains 0.55. We will check the probability of making 4 consecutive shots to see if it is so low that it would suggest a change in ability. The null hypothesis is that his shooting percentage has not changed. The alternate hypothesis is that his shooting percentage has increased.

Calculate the Probability of the Streak

If the player takes his free throw shots independently, the probability of making 4 consecutive shots (4 successful events in a row) is calculated as: \( 0.55^4 \).

Compute the Value

Calculate the value using the probability: \( 0.55^4 = 0.0915 \). This means the probability of making 4 shots in a row is about 9.15%.

Evaluate the Evidence

A 9.15% probability of making 4 shots in a row, while not very high, is not exceptionally rare or unusual. Generally, a probability threshold (such as 5% or lower) is used to determine if an event is statistically significant.

Conclusion

Since 9.15% is higher than typical significance thresholds, this streak does not provide strong evidence of a change in the player's probability to make shots. Rather, it falls within the expected variability of his performance.

Key concepts

Null Hypothesis

In statistics, we often start by assuming that nothing has changed. This is known as the null hypothesis. It is a default position that suggests there is no effect or no difference. In our basketball example, the null hypothesis states that the player's skill level has not improved, despite making four consecutive shots. His success rate remains at 55%, as it usually is. Understanding the null hypothesis helps set a clear starting point for analysis. If evidence is to suggest otherwise, this hypothesis must be statistically challenged and potentially rejected. It's like assuming every coin you flip is fair unless proven otherwise. This approach helps to avoid premature conclusions about changes in skill or other phenomena. The opposite of the null hypothesis is the alternative hypothesis, which suggests that the embedded pattern observed (like the streak in shots) is due to a change in skill or probability of success. Evaluating these hypotheses will require further statistical methods to determine whether the evidence against the null is strong enough to make a change.

Statistical Significance

Statistical significance is a tool we use in statistics to decide whether our observations are meaningful. The term itself refers to the likelihood that our results are not just due to random chance. For our basketball player's case, statistical significance would help determine if making 4 shots in a row is simply luck or a sign of improved skill. To assess this, we look at certain thresholds known as significance levels, typically set at 5% (0.05). This means that if the likelihood of the observed event (or something more extreme) happening is less than 5%, we might consider the event statistically significant. Applying this concept, the player's 9.15% probability of making four consecutive shots doesn't cross the typical threshold of 5%. This implies the event isn't statistically significant enough to claim a change in his probability of success. In essence, while making 4 shots might be noteworthy, it doesn't necessarily indicate a shift in ability.

Probability Calculations

Probability calculations involve using numbers to describe the likelihood of an event happening. For the basketball player, we calculated the probability of hitting four consecutive shots based on his usual success rate.Using the formula for independent events, the probability of making a single shot is 0.55 or 55%. When calculating the chance of making four shots in a row, we multiply his success rate for each shot:\[0.55 \times 0.55 \times 0.55 \times 0.55 = 0.55^4 = 0.0915\]This results in about a 9.15% chance of him achieving this streak. Each shot is considered independently, assuming his performance doesn’t influence subsequent attempts. These calculations help in understanding whether certain results (like the 4-success streak) are to be expected from random variability or possibly indicative of improved player performance. It offers a mathematical way to weigh "luck" against "actual performance changes."