Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.1.2

Chapter 6, Problem 6.1.2

Which statistic is the best unbiased estimator for μ?
a. s
b. xbar
c. the median
d. the mode

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Understand the concept of an unbiased estimator: An unbiased estimator is a statistic whose expected value is equal to the parameter it is estimating. In this case, we are looking for the best unbiased estimator for the population mean (μ).

Review the options provided: (a) s (sample standard deviation), (b) x̄ (sample mean), (c) the median, and (d) the mode. Consider which of these statistics is most directly related to estimating the population mean.

Recall that the sample mean (x̄) is an unbiased estimator of the population mean (μ). This means that the expected value of x̄ is equal to μ, making it the best choice among the options.

Understand why the other options are not the best unbiased estimators: (a) The sample standard deviation (s) estimates the population standard deviation, not the mean. (c) The median and (d) the mode are measures of central tendency but are not guaranteed to be unbiased estimators of the population mean.

Conclude that the sample mean (x̄) is the best unbiased estimator for μ because it satisfies the condition of unbiasedness and is specifically designed to estimate the population mean.

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Key Concepts

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

Unbiased Estimator

An unbiased estimator is a statistical estimator that, on average, equals the parameter it estimates. This means that if you were to take many samples and calculate the estimator for each sample, the average of those estimates would converge to the true parameter value. In this context, we are looking for an estimator of the population mean (μ) that does not systematically overestimate or underestimate it.

Sample Mean (x̄)

The sample mean, denoted as x̄, is the average of a set of sample observations. It is calculated by summing all the sample values and dividing by the number of observations. The sample mean is a commonly used estimator for the population mean (μ) and is considered an unbiased estimator, making it a strong candidate in the context of the question.

Other Measures of Central Tendency

Measures of central tendency, such as the median and mode, summarize a set of data points. The median is the middle value when data is ordered, while the mode is the most frequently occurring value. Although these measures provide insights into the data, they are not unbiased estimators of the population mean (μ) like the sample mean (x̄), which is why they are less suitable in this context.

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Related Practice

Textbook Question

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

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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.

117

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Textbook Question

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

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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

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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

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Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.

μ<33

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