Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.Tech.1a

Chapter 6, Problem 6.Tech.1a

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.

Use technology to find a 95% confidence interval for the population proportion of adults who
a. view foreign trade as an economic opportunity.

Verified step by step guidance

Step 1: Identify the sample proportion (p̂) for adults who view foreign trade as an economic opportunity. From the table, 63% of the 1021 respondents view foreign trade as an opportunity. Convert this percentage to a proportion: p̂ = 0.63.

Step 2: Determine the sample size (n). The problem states that 1021 U.S. adults were polled, so n = 1021.

Step 3: Calculate the standard error (SE) for the population proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n). Substitute p̂ = 0.63 and n = 1021 into the formula.

Step 4: Find the critical value (z*) for a 95% confidence interval. For a 95% confidence level, the z* value is approximately 1.96.

Step 5: Compute the margin of error (ME) using the formula: ME = z* * SE. Then, construct the confidence interval using the formula: Confidence Interval = p̂ ± ME.

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Key Concepts

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

Population Proportion

Population proportion refers to the fraction of a population that exhibits a certain characteristic. In the context of the Gallup poll, it represents the percentage of U.S. adults who view foreign trade as an economic opportunity. Understanding population proportions is crucial for interpreting survey results and making inferences about the broader population based on sample data.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, such as 95%. For the Gallup poll, calculating a 95% confidence interval for the proportion of adults viewing foreign trade as an opportunity allows researchers to estimate the uncertainty around the sample proportion and assess the reliability of the results.

Sampling Error

Sampling error is the difference between the sample statistic and the actual population parameter due to the fact that only a subset of the population is surveyed. In the context of the Gallup poll, it reflects the potential inaccuracies in the reported percentages of adults who view foreign trade as an opportunity or threat. Understanding sampling error is essential for evaluating the precision of survey results and the validity of conclusions drawn from them.

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Sampling Distribution of Sample Proportion

Related Practice

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.

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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

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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.

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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.

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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.

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Textbook Question

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

d. Determine the minimum sample size required to be 95% confident that the sample mean test score is within 10 points of the population mean test score.

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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.

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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.

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Textbook Question

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

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