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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Related Practice

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.

Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

Textbook Question

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

Textbook Question

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

a. Find the point estimate for the population proportion.

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^?

Textbook Question

Annual Precipitation The annual precipitation amounts (in inches) of a random sample of 61 years for Chicago, Illinois, have a sample standard deviation of 6.46. Use a 98% level of confidence. (Source: National Oceanic and Atmospheric Administration)