Question

A media report claims that 50% of U.S. teens with smartphones feel addicted to their devices. A skeptical researcher believes that this figure is too high. She decides to test the claim by taking a random sample of 100 U.S. teens who have smartphones. Only 40 of the teens in the sample feel addicted to their devices. Does this result give convincing evidence that the media report’s 50% claim is too high? To find out, we want to perform a simulation to estimate the probability of getting 40 or fewer teens who feel addicted to their devices in a random sample of size 100 from a very large population of teens with smartphones in which 50% feel addicted to their devices.

Let 1 = feels addicted and 2 = doesn’t feel addicted. Use a random number generator to produce 100 random integers from 1 to 2. Record the number of 1’s in the simulated random sample. Repeat this process many, many times. Find the percent of trials on which the number of 1’s was 40 or less.

Does the problem describe a valid simulation design? Justify your answer.

Answer 1

The simulation design described in the problem is valid for estimating the probability of finding 40 or fewer teens who feel addicted to their devices in a sample of 100 when the population addicted rate is 50%.

Step-by-step explanation:

The problem describes a valid simulation design to estimate the probability of getting 40 or fewer teens who feel addicted to their devices in a random sample of size 100 when 50% of a larger population feels addicted. This simulation imitates the process of randomly selecting teens and assessing their feeling of addiction. In the simulation, '1' represents a teen who feels addicted, and '2' represents a teen who does not. By using a random number generator to produce 100 integers ranging from 1 to 2 and recording the number of 1’s, we simulate the scenario of teens feeling addicted to their devices. The process is repeated many times to approximate the probability of observing 40 or fewer teens who feel addicted, thus assessing the claim made in the media report. This simulated approach is a common statistical method called a proportion simulation.

Answer 2

The simulation outlined is a valid approach to test the skeptical researcher's belief against the media report's claim that 50% of U.S. teens with smartphones feel addicted by simulating a binomial process reflecting the conditions of the claim.

Step-by-step explanation:

The scenario described represents a classic binomial probability situation, where there are only two outcomes (addicted: 1, or not addicted: 2) for each trial (each teen). For a valid simulation, we should generate a large number of samples of 100 random integers where each integer has a 50% chance of being a 1 (feels addicted) or a 2 (doesn’t feel addicted).

We then count the number of 1's in each sample, and calculate the proportion of these samples that have 40 or fewer 1's. If this proportion is small, it indicates that the result of 40 teens feeling addicted in a sample of 100, while possible under the media report's claim, is unlikely, and would suggest that the actual percentage may be less than 50%. However, if the proportion is not small, the evidence is not strong enough to suggest that the media claim is too high. The simulation is valid because it imitates the random sampling process under the conditions specified in the media report (that 50% of teens feel addicted to their smartphones).