Predicting Samples and Populations Using Proportions

Ahmad saw a report that claimed 57% of US adults think a third major political party is needed. He was curious how students at his large university felt on the topic, so he asked the same question to a random sample of 100 students and made a 95% confidence interval to estimate the proportion of students who agreed that a third major political party was needed. His resulting interval was (0.599, 0.781) Assume that the conditions for inference were all met. Based on his interval, is it plausible that 57% of all students at his university would agree that a third party is needed?

• Yes, 57% is close to the interval, so if you expanded an additional standard deviation away, 57% would be in the interval.

• No, 57% is not in the interval of plausible values.

• Yes, because the true proportion at this university is likely higher.

• No, because the true proportion at this university is likely lower.

Ahmad's sister, Diedra, was curious how students at her large high school would answer the question, "Do we need a third major political party?", so she asked it to a random sample of 100 students at her school. She also made a 95% confidence interval to estimate the proportion of students at her school who would agree that a third party is needed. Her interval was (0.557,0.743). Assume that the conditions for inference were all met. Based on her interval, is it plausible that 57% of students at her school would agree that a third party is needed?

• No, 57% is not the point estimate.

• Yes, 57% is in the interval of plausible values for the population proportion.

• Yes, because the true proportion at this school is likely higher.

• No, because the true proportion at this school is likely lower.

A video game gives players a reward of gold coins after they defeat an enemy. The creators of the game want players to have a chance at earning bonus coins when they defeat a certain challenging enemy. The creators attempt to program the game so that the bonus is awarded randomly with a 30% probability after the enemy is defeated. To see if the bonus is being awarded as intended, the creators defeated the enemy in a series of 100 attempts (they're willing to treat this as a random sample). After each attempt, they recorded whether or not the bonus was awarded. They used the results to build a 95% confidence interval for p, the proportion of attempts that will be rewarded with the bonus. The resulting interval was (0.323, 0.517).

• The bonus most likely isn't being awarded at the intended rate of 30%.

• It's plausible that the bonus is being awarded at the intended rate of 30%.

• The bonus is being awarded at the intended rate of 30%.

The creators of a video game want players to have a chance at earning a rare item when they defeat a challenging enemy. The creators attempt to program the game so that the rare item is awarded randomly with a 15% probability after the enemy is defeated. To see if the rare item is being awarded as intended, the creators defeated the enemy in a series of 100 attempts (they're willing to treat this as a random sample). After each attempt, they recorded whether or not the rare item was awarded. They used the results to build a 95% confidence interval for p, the proportion of attempts that will be rewarded with the rare item, which was 0.12±0.06. What does this interval suggest?

• The rare item most likely isn't being awarded at the intended rate of 15%.

• The rare item is being awarded at the intended rate of 15%.

• It's plausible that the rare item is being awarded at the intended rate of 15%

A certain flight transports over 1,000 passengers daily, and the airline is curious what proportion of passengers on this flight don't use overhead bin space. They take an SRS of 40 of these passengers and find that 6 of them don't use overhead bin space. The airline wants to use this data to create a one-sample z-interval to estimate the proportion of passengers on this flight don't use overhead bin space. Which conditions for constructing this confidence interval did the sample meet?
Choose all answers that apply:

• The data is from a random sample from the population of interest.

• Individual observations can be considered independent.

A market analyst is curious what proportion of Los Angeles residents have a landline telephone. A survey of 200 randomly selected Los Angeles residents shows that 96 of those selected have a landline telephone. The analyst wants to use this data to construct a one-sample z-interval for a proportion. Based on this sample, which of the following is a 90% confidence interval for the proportion of Los Angeles residents who have a landline telephone?

• 0.48±0.069

• 0.48±0.058

• 96±0.069

• 96±0.058

What is the critical value z∗, for constructing a 92%

confidence interval?

A large shipping company wants each order to arrive within the delivery window they give to the customer. They plan on taking a random sample of orders to construct a one-sample z-interval to estimate what proportion of all orders arrive within their delivery window. They'll use a confidence level of 95%, and they don't want the margin of error to exceed 2 percentage points. Previous data suggests that about 90% of orders arrive within their delivery window. If we assume p-hat=0.90​, what is the smallest sample size required to obtain the desired margin of error?

• 366

• 588

• 865

• 1116

A survey of 400 randomly selected homes from a large city with over 30,000 homes showed that 16 of the sampled homes didn't have a television. Based on this sample, which of the following is a 95% confidence interval for the proportion of homes in this city that don't have a television?