Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

문제 7.1.22

7장, 문제 7.1.22

"Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Internet Use A random sample of 5005 adults in the United States includes 751 who do not use the Internet (based on three Pew Research Center polls). Construct a 95% confidence interval estimate of the percentage of U.S. adults who do not use the Internet. Based on the result, does it appear that the percentage of U.S. adults who do not use the Internet is different from 48%, which was the percentage in the year 2000?"

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Step 1: Identify the sample proportion (p̂). The sample proportion is calculated as the number of successes (in this case, adults who do not use the Internet) divided by the total sample size. Use the formula: p̂ = x / n, where x = 751 and n = 5005.

Step 2: Calculate the standard error (SE) for the sample proportion. The formula for the standard error is: SE = sqrt((p̂ * (1 - p̂)) / n). Substitute the value of p̂ from Step 1 and the sample size n = 5005 into this formula.

Step 3: Determine the critical value (z*) for a 95% confidence level. For a 95% confidence level, the critical value z* is approximately 1.96 (based on the standard normal distribution).

Step 4: Construct the confidence interval for the population proportion (p). Use the formula: Confidence Interval = p̂ ± z* × SE. Substitute the values of p̂, z*, and SE from the previous steps into this formula to calculate the lower and upper bounds of the confidence interval.

Step 5: Compare the confidence interval to the value 48% (0.48). Check if 0.48 lies within the confidence interval. If it does not, conclude that the percentage of U.S. adults who do not use the Internet is significantly different from 48%. If it does, conclude that there is no significant difference.

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주요 개념

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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. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population proportion.

Sample Proportion

The sample proportion is the ratio of the number of successes (in this case, adults who do not use the Internet) to the total number of observations in the sample. It is calculated as p̂ = x/n, where x is the number of successes and n is the sample size. This value is crucial for estimating the population proportion and constructing the confidence interval.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this context, it involves comparing the constructed confidence interval to the historical percentage of 48% to determine if there is a statistically significant difference in the proportion of U.S. adults who do not use the Internet compared to the year 2000.

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