Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.Tech.3b

Chapter 6, Problem 6.Tech.3b

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

b. What was the greatest value you obtained for p^?

Verified step by step guidance

Step 1: Understand the problem. The question asks for the greatest value of p^ (sample proportion) obtained from the Gallup World Affairs Poll of 2021. The image provides parameters for a binomial distribution simulation, which can help analyze probabilities or proportions.

Step 2: Recall the formula for the sample proportion p^: p^ = x/n, where x is the number of successes and n is the total number of trials. In this case, n = 1021 (the number of U.S. adults polled).

Step 3: Analyze the image. The Random Number Generation tool is set to simulate a binomial distribution with p = 0.54 (probability of success) and 1537 trials. However, this simulation is unrelated to the actual poll data, which involves 1021 adults. Focus on the poll data for calculating p^.

Step 4: To find the greatest value of p^, identify the category with the highest number of successes (x) from the poll data. If the data is not explicitly provided, assume the highest percentage corresponds to the greatest p^.

Step 5: Calculate the greatest p^ using the formula p^ = x/n, where x corresponds to the highest percentage of responses multiplied by n (1021). Ensure to convert the percentage into a decimal before multiplying.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Proportion Estimate (p^)

The proportion estimate, denoted as p^ (p-hat), represents the sample proportion of a certain characteristic in a population. It is calculated by dividing the number of successes (individuals with the characteristic) by the total number of observations in the sample. In the context of polling, p^ indicates the percentage of respondents who expressed a particular opinion.

Binomial Distribution

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). This distribution is useful for modeling scenarios like polling, where outcomes can be classified as success or failure.

Random Sampling

Random sampling is a technique used to select a subset of individuals from a larger population, ensuring that each individual has an equal chance of being chosen. This method helps to obtain a representative sample, which is crucial for making valid inferences about the population. In the context of the Gallup poll, random sampling ensures that the opinions of the 1021 U.S. adults reflect the broader population's views.

Recommended video:

04:17

Simple Random Sampling

Related Practice

Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

views

Textbook Question

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

[IMAGE]

Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

[IMAGE]

Use the finite population correction factor to construct each confidence interval for the population mean.

a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

197

views

Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

views

Textbook Question

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

views

Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

[IMAGE]

Find the minimum sample size needed to estimate, with 95% confidence, the population proportion of adults who feel that China’s economic power is a critical or an important economic threat to the United States. Your estimate must be accurate within 2% of the population proportion.

views

Textbook Question

Use technology to find a 95% confidence interval for the population proportion of adults who

a. view foreign trade as an economic opportunity.

views