Question

The following questions are from ARTIST (reproduced with permission) Consider the question of whether the home team wins more than half of its games in the National Basketball Association. Suppose that you study a simple random sample of 80 professional basketball games and find that 52 of them are won by the home team. a. Assuming that there is no home court advantage and that the home team therefore wins \(50 \%\) of its games in the long run, determine the probability that the home team would win \(65 \%\) or more of its games in a simple random sample of 80 games. b. Does the sample information (that 52 of a random sample of 80 games are won by the home team) provide strong evidence that the home team wins more than half of its games in the long run? Explain.

Short answer

The sample suggests strong evidence the home team wins more than half its games.

Step-by-step solution

Define the Null and Alternative Hypotheses

The null hypothesis \(H_0\) is that the home team wins 50% of its games: \(p = 0.5\). The alternative hypothesis \(H_a\) is that the home team wins more than 50% of its games: \(p > 0.5\).

Calculate the Sample Proportion

Given that 52 out of 80 games are won by the home team, the sample proportion \(\hat{p}\) is calculated as:\[ \hat{p} = \frac{52}{80} = 0.65 \]

Compute the Standard Error (SE)

The standard error for the proportion is calculated using the null hypothesis \(p = 0.5\): \[ SE = \sqrt{\frac{0.5 \times (1 - 0.5)}{80}} = \sqrt{\frac{0.25}{80}} = \sqrt{0.003125} \approx 0.0559 \]

Find the Z-score

The Z-score, which measures how many standard errors the sample proportion is from the null hypothesis proportion, is calculated as:\[ Z = \frac{\hat{p} - p}{SE} = \frac{0.65 - 0.5}{0.0559} \approx 2.68 \]

Determine the P-value

Use the Z-score to find the p-value from the standard normal distribution. A Z-score of 2.68 corresponds to a p-value of approximately 0.0037, which indicates the probability of observing a sample proportion of 0.65 or more assuming \(p = 0.5\).

Make the Decision

Because the p-value (0.0037) is less than the typical significance level (e.g., 0.05), we reject the null hypothesis. This suggests that there is strong evidence that the home team wins more than half of its games in the long run.

Key concepts

Null and Alternative Hypotheses

In hypothesis testing, we start by defining what we believe to be true, known as the null hypothesis, and what we suspect might be true, known as the alternative hypothesis.

• The **null hypothesis** ( \(H_0\)) in our context means that the home team wins 50% of their games. Mathematically, this is expressed as \(p = 0.5\), where \(p\) is the probability.
• The **alternative hypothesis** ( \(H_a\)) is what we think is true instead, which in this case is that the home team wins more than 50% of its games, thus \(p > 0.5\).

In hypothesis testing, we assume the null hypothesis is true and then check the evidence provided by the sample to accept or reject it.

Sample Proportion

The sample proportion is a critical concept in statistics as it gives us an estimate of the true proportion based on our sample data. Here, we found that 52 out of the 80 games were won by the home team. To calculate the sample proportion (\(\hat{p}\)):- **Formula**: \(\hat{p} = \frac{52}{80}\)- **Calculation**: This results in \(\hat{p} = 0.65\)This means that according to our sample, 65% of games are won by the home team. Through this, we gain an initial idea that the proportion of home wins might be higher than what the null hypothesis suggests.

Standard Error

The standard error provides us with a measure of the sampling variability. - **Definition**: It shows how much the sample proportion \(\hat{p}\) would differ from the true population proportion just by chance.- **Formula**: For proportions, the standard error ( \(SE\)) is calculated as: \[SE = \sqrt{\frac{p(1-p)}{n}}\] Substituting in our values using the null hypothesis proportion \(p = 0.5\), and the sample size \(n = 80\), we get: \[SE = \sqrt{\frac{0.5 \times 0.5}{80}} \approx 0.0559\]This value (0.0559) tells us how much variation we might expect between the sample and the population proportion, assuming the null hypothesis is correct.

Z-score

The Z-score is a measure that tells us how far away our sample statistic is from the population parameter, measured in standard errors.- **Calculation**: We subtract the population proportion from our sample proportion and divide that by the standard error: \[Z = \frac{\hat{p} - p}{SE}\] Substituting the numbers: \[Z = \frac{0.65 - 0.5}{0.0559} \approx 2.68\]This Z-score of 2.68 shows that our sample proportion is 2.68 standard errors above the null hypothesis proportion of 0.5. In simpler terms, this tells us that our sample result is quite far from what the null hypothesis would predict.

P-value

The p-value helps us determine the strength of the evidence against the null hypothesis. - **Definition**: It is the probability of getting a sample result as extreme as the one observed, assuming that the null hypothesis is true. - **Calculation**: We use the Z-score to find this probability from the standard normal distribution. With a Z-score of 2.68, the p-value is approximately 0.0037. Since this p-value is lower than the common significance level of 0.05, it suggests that the observed data would be very unlikely if the null hypothesis were true. Thus, we reject the null hypothesis and conclude there is strong evidence that the true proportion of home wins is greater than 50%.