Question

a worker at the local department of motor vehicles (dmv) claims that 60% of teenagers smile in their drivers license photo. in a random sample of 10 teenagers from last months new drivers licenses, only 4 of them were smiling in their photos. to see how unusual this sample is, 100 simulated trials were conducted under the assumption that 60% of teenagers smile for their drivers license photo. based on the dotplot of the simulation results and the random sample from last months new drivers licenses, which conclusion can be drawn? the actual proportion of teenagers who do not smile is only 8%. it is clear that exactly 6 out of 10 teenagers will smile in their drivers license photo. if we continued to take more samples of 10 teenagers, the center of the distribution would shift to 6. there is about an 8% chance that 4 or fewer teenagers smiled for their photo. this is not unusual and is not convincing evidence that the true probability is less than 60%.

Explanation:

1. Analyze Option 1: The dotplot and simulation are about smiling teenagers, not directly about non - smiling proportion calculation. So this option is incorrect.
2. Analyze Option 2: The claim is a proportion (60% smile), not that exactly 6 out of 10 will smile. Random samples have variability, so this is wrong.
3. Analyze Option 3: The center of the distribution (expected value) for a binomial distribution with \(n = 10\) and \(p=0.6\) is \(np = 6\). But taking more samples won't shift the center; the center is determined by the population proportion. So this is incorrect.
4. Analyze Option 4: From the simulation (100 trials), if about 8% of trials had 4 or fewer smiling, this means the chance of getting 4 or fewer is about 8%. Since 8% is not extremely low, it's not unusual and not strong evidence that the true probability is less than 60%. This matches the logic of simulation - based inference.

Answer:

There is about an 8% chance that 4 or fewer teenagers smiled for their photo. This is not unusual and is not convincing evidence that the true probability is less than 60%.