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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 6 - Confidence Intervals문제 6.R.21
6장, 문제 6.R.21

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.
In a survey of 73,901 college graduates, 23,991 obtained a postgraduate degree. (Adapted from Gallup)

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Step 1: Calculate the point estimates for p and q. The population proportion p is estimated as the ratio of the number of successes (graduates with postgraduate degrees) to the total number of observations (total graduates). Use the formula p = x / n, where x = 23,991 and n = 73,901. Then, calculate q as q = 1 - p.
Step 2: Construct the 90% confidence interval for p. Use the formula for a confidence interval: CI = p ± Z * sqrt((p * q) / n), where Z is the critical value for the desired confidence level (for 90%, Z ≈ 1.645). Substitute the values of p, q, and n into the formula to calculate the margin of error and the interval.
Step 3: Construct the 95% confidence interval for p. Use the same formula as in Step 2, but with the critical value Z ≈ 1.96 for a 95% confidence level. Again, substitute the values of p, q, and n to calculate the margin of error and the interval.
Step 4: Interpret the results of the confidence intervals. Explain that the 90% confidence interval means we are 90% confident that the true population proportion p lies within the interval, and similarly, the 95% confidence interval means we are 95% confident that the true population proportion p lies within the interval.
Step 5: Compare the widths of the confidence intervals. Note that the 95% confidence interval is wider than the 90% confidence interval because a higher confidence level requires a larger margin of error to ensure the true population proportion is captured within the interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Point Estimate

A point estimate is a single value that serves as an approximation of a population parameter. In this context, the point estimate for the population proportion p is calculated by dividing the number of individuals with a postgraduate degree by the total number of surveyed individuals. This provides a quick snapshot of the proportion of graduates who pursued further education.
추천 영상:
가이드 코스
06:33
Introduction to Confidence Intervals

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 90% confidence interval suggests that if we were to take many samples, approximately 90% of the calculated intervals would contain the true population proportion p. This concept is crucial for understanding the precision and reliability of the point estimate.
추천 영상:
가이드 코스
06:33
Introduction to Confidence Intervals

Width of Confidence Intervals

The width of a confidence interval reflects the level of uncertainty associated with the estimate of the population parameter. A wider interval indicates greater uncertainty, while a narrower interval suggests more precision. Comparing the widths of 90% and 95% confidence intervals helps to illustrate the trade-off between confidence level and precision, as higher confidence levels typically result in wider intervals.
추천 영상:
가이드 코스
06:33
Introduction to Confidence Intervals
관련 실천
교과서 질문

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.98, s = 0.9, n = 12, xbar = 6.8

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교과서 질문

[APPLET] The waking times (in minutes past 5:00 A.M.) of 40 people who start work at 8:00 A.M. are shown in the table at the left. Assume the population standard deviation is 45 minutes. Find (a) the point estimate of the population mean μ and (b) the margin of error for a 90% confidence interval.

85
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교과서 질문

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.99, n = 10

97
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교과서 질문

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.90, n = 16

74
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교과서 질문

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.90, s = 25.6, n = 16, xbar = 72.1

148
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교과서 질문

In Exercise 19, would it be unusual for the population proportion to be 38%? Explain.

90
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