Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.1.9

Chapter 6, Problem 6.1.9

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 (denoted as x̄) and the population mean (denoted as μ). It is calculated as: Sampling Error = x̄ - μ.

Step 2: Identify the values provided in the problem. From the number line, the sample mean x̄ is given as 3.8, and the population mean μ is given as 4.27.

Step 3: Write the formula for sampling error using MathML:

Sampling Error = x̄ - μ

Step 4: Substitute the values into the formula. Replace x̄ with 3.8 and μ with 4.27 in the formula.

Step 5: Perform the subtraction to find the sampling error. The result will be the difference between 3.8 and 4.27, which represents the sampling error.

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Key Concepts

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

Sampling Error

Sampling error refers to the difference between the sample mean (x̄) and the population mean (μ). It occurs because a sample is only a subset of the population, and thus may not perfectly represent the entire population. In this case, the sampling error can be calculated by subtracting the population mean from the sample mean, providing insight into the accuracy of the sample's estimate.

Sample Mean (x̄)

The sample mean, denoted as x̄, is the average value of a set of observations drawn from a population. It is calculated by summing all the sample values and dividing by the number of observations. In the context of the question, the sample mean is given as 3.8, which serves as a point of comparison against the population mean to assess sampling error.

Population Mean (μ)

The population mean, represented by μ, is the average of all possible values in a population. It is a fixed value that describes the entire group being studied. In this scenario, the population mean is 4.27, and understanding its relationship to the sample mean is crucial for evaluating the sampling error and the representativeness of the sample.

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Population Standard Deviation Known

Related Practice

Textbook Question

In Exercise 35, would it be unusual for the population mean to be over \$1500? Explain.

Textbook Question

Tennis Ball Manufacturing A company manufactures tennis balls. When the balls are dropped onto a concrete surface from a height of 100 inches, the company wants the mean bounce height to be 55.5 inches. This average is maintained by periodically testing random samples of 25 tennis balls. If the t-value falls between and , then the company will be satisfied that it is manufacturing acceptable tennis balls. For a random sample, the mean bounce height of the sample is 56.0 inches and the standard deviation is 0.25 inch. Assume the bounce heights are approximately normally distributed. Is the company making acceptable tennis balls? Explain.

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Textbook Question

Which statistic is the best unbiased estimator for μ?

a. s

b. xbar

c. the median

d. the mode

Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.99, s^2 = 0.64, n = 7

Textbook Question

In Exercises 7 and 8, find the margin of error for the values of c, s, and n.

c = 0.99, s = 3, n = 6

Textbook Question

In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.

c = 0.97