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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
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.3.1
Chapter 6, Problem 6.3.1

True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
The point estimate for the population proportion of failures is 1-p^

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1
Understand the problem: The statement is about the point estimate for the population proportion of failures. Recall that the population proportion of successes is denoted by p, and the population proportion of failures is complementary to this, represented as 1 - p.
Review the concept: The point estimate for the population proportion of failures is derived from the complement of the population proportion of successes. If p represents the proportion of successes, then 1 - p represents the proportion of failures.
Analyze the statement: The statement claims that the point estimate for the population proportion of failures is 1 - p^ (where p^ is the sample proportion of successes). This aligns with the definition of the complement rule in probability.
Determine the truth value: Since the statement correctly describes the relationship between the sample proportion of successes (p^) and the sample proportion of failures (1 - p^), the statement is true.
Conclude: The statement is true, and no rewriting is necessary. If the statement were false, you would need to correct it by ensuring the complement relationship is properly stated as 1 - p^ for the sample proportion of failures.

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Key Concepts

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

Point Estimate

A point estimate is a single value that serves as an approximation of a population parameter. In statistics, it is often derived from sample data and is used to infer characteristics about the entire population. For example, the sample mean is a point estimate of the population mean.
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Population Proportion

The population proportion refers to the fraction of a population that possesses a certain characteristic. It is denoted by 'p' and is crucial in understanding the distribution of categorical data. For instance, if 30 out of 100 surveyed individuals prefer a certain product, the population proportion of preference is 0.3.
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Complement of a Proportion

The complement of a proportion is the probability that an event does not occur, calculated as 1 minus the proportion of the event. In the context of failures, if 'p' represents the proportion of successes, then '1 - p' represents the proportion of failures. This concept is essential for accurately interpreting and calculating probabilities in statistical analysis.
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Related Practice
Textbook Question

In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.

(0.512, 0.596)

Textbook Question

"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.

Social Security In a survey of 351 retired Americans, 200 said that they rely on Social Security as major source of income. (Adapted from Gallup)"

Textbook Question

In Exercises 13–16, find the margin of error for the values of c, σ and n.

c = 0.975, σ = 4.6, n = 100

Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.80, σ = 4.1, E = 2.

Textbook Question

In Exercise 31, the population mean salary is \$67,319. Does the t-value fall between -t0.98 and t0.98? (Source: Salary.com)

Textbook Question

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.98