Question

Performers in the Rock and Roll Hall of Fame From its founding through \(2015,\) the Rock and Roll Hall of Fame has inducted 303 groups or individuals, and 206 of the inductees have been performers while the rest have been related to the world of music in some way other than as a performer. The full dataset is available in RockandRoll. (a) What proportion of inductees have been performers? Use the correct notation with your answer. (b) If we took many samples of size 50 from the population of all inductees and recorded the proportion who were performers for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

Short answer

For part (a), the proportion of inductees that have been performers is \( \frac{206}{303} \) . For part (b), the distribution of the sample proportions is expected to be approximately normal (given the sample size is sufficiently large), and it would be centered around this proportion value, \( \frac{206}{303} \) .

Step-by-step solution

Proportion Calculation

The proportion of inductees that have been performers can be calculated by dividing the number of performers by the total number of inductees: \( P = \frac{n_{performers}}{n_{total}} \) where \( n_{performers} = 206 \) and \( n_{total} = 303 \). This will give us the proportion of performers.

Calculate Proportion Value

Plugging the values into the equation from Step 1, we get: \( P =\frac{206}{303} \)

Understanding the Distribution of Sample Proportions

According to the Central Limit Theorem, if we took many samples of size 50 from the total population and recorded the proportion who were performers each time, the shape of the distribution of these sample proportions would be approximately normal. This is because the sample size is sufficiently large (greater than 30). The center or mean of this distribution would be expected to be around the same as the population proportion, calculated in Step 2.