Question

A survey of 2,250 adults reported that 59% watch news videos. Complete parts (a) through (c) below.

a. Suppose that you take a sample of 100 adults. If the population proportion of adults who watch news videos is 0.59, what is the probability that fewer than half in your sample will watch news videos? Round to four decimal places as needed.

b. Suppose that you take a sample of 500 adults. If the population proportion of adults who watch news videos is 0.59, what is the probability that fewer than half in your sample will watch news videos? Round to four decimal places as needed.

c. Discuss the effect of sample size on the sampling distribution of the proportion in general and the effect on the probabilities in parts (a) and (b). Choose the correct answer below.

A. Increasing the sample size by a factor of 5 increases the standard error by a factor of √5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.59 and increases the probability in part (b).

B. Increasing the sample size by a factor of 5 decreases the standard error by a factor of √5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.59 and decreases the probability in part (b).

C. The probabilities in parts (a) and (b) are the same. Increasing the sample size does not change the sampling distribution of the proportion

Answer 1

The student's question pertains to calculating probabilities for a population proportion and the impact of sample size on the precision of the sampling distribution. Larger sample sizes result in lower standard error and more concentration around the true proportion.

Step-by-step explanation:

The subject of the student's question involves calculating probabilities related to the proportion of a population engaging in a specific activity (watching news videos) and understanding the effects of sample size on the sampling distribution of the proportion. This applies to the field of statistics, particularly in the context of inferential statistics where the central limit theorem plays a crucial role.

To calculate the probability of an event when dealing with a population proportion, students often employ the normal approximation to the binomial distribution. However, part (c) of the question specifically addresses the impact of changing sample sizes on the standard error and the concentration of the sampling distribution around the population proportion. In general, as the sample size increases, the standard error decreases, thus making the sampling distribution more concentrated around the true population proportion.

For the question's parts (a) and (b), we use normal approximation to find the probabilities of fewer than half of the samples watching news videos. The sample size does indeed affect the probability. Larger sample sizes lead to a lower standard error and, typically, a smaller probability of finding a sample proportion that significantly deviates from the population proportion.