Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.1.10

Chapter 6, Problem 6.1.10

Graphical Analysis In Exercises 9–12, use the values on the number line to find the sampling error.

Verified step by step guidance

Step 1: Understand the concept of sampling error. Sampling error is the difference between the sample mean (x̄) and the population mean (μ). It is calculated as:

Sampling error = x̄ - μ

Step 2: Identify the values provided in the problem. From the number line, the population mean (μ) is given as 8.76, and the sample mean (x̄) is given as 9.5.

Step 3: Substitute the values into the formula for sampling error. Replace μ with 8.76 and x̄ with 9.5 in the formula:

Sampling error = 9.5 - 8.76

Step 4: Perform the subtraction operation to find the sampling error. This step involves calculating the difference between the sample mean and the population mean.

Step 5: Interpret the result. The sampling error represents how much the sample mean deviates from the population mean, which can provide insights into the accuracy of the sample in representing the population.

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Key Concepts

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

Population Mean (μ)

The population mean, denoted as μ, is the average of all values in a population. It serves as a central point around which data points are distributed. In the context of the question, μ is given as 8.76, indicating the expected average value of the entire population from which a sample is drawn.

Sample Mean (x̄)

The sample mean, represented as x̄, is the average of values in a sample taken from the population. It provides an estimate of the population mean based on the data collected. In this case, x̄ is 9.5, which suggests that the sample's average is higher than the population mean, indicating potential sampling variability.

Sampling Error

Sampling error refers to the difference between the sample mean (x̄) and the population mean (μ). It quantifies how much the sample mean deviates from the true population mean due to random sampling. In this scenario, the sampling error can be calculated as x̄ - μ, which helps assess the accuracy of the sample in representing the population.

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Related Practice

Textbook Question

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (c) construct a 98% confidence interval for the population mean.

[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

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Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.98, n = 26

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Textbook Question

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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Textbook Question

What happens to the shape of the chi-square distribution as the degrees of freedom increase?

222

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Textbook Question

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 2094 U.S. adults who have used an online dating app, 57% said their personal experience with online dating was positive. The survey’s margin of error is ±3.6%. (Source: Pew Research Center)

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Textbook Question

For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.

a. 90%

b. 95%

c. 98%

d. 99%

125

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