Chapter 7: Q.20 (page 431)

A sample of teens A study of the health of teenagers plans to measure the blood cholesterol levels of an SRS of $13$ - to $16$ -year-olds. The researchers will report the mean $x$ from their sample as an estimate of the mean cholesterol level M in this population.
(a) Explain to someone who knows no statistics what it means to say that $x$ is an unbiased estimator of $μ$ .
(b) The sample result x is an unbiased estimator of the population mean $μ$ no matter what size SRS the study chooses. Explain to someone who knows no statistics why a large random sample gives more trustworthy results than a small random sample.

Expert verified

a). The average of all possible $\bar{x}$ -values is equal to $μ$ .

b). A larger sample gives more information and less variability.

Step by step solution

A study of the health of teenagers plans to measure the blood cholesterol levels of an SRS of $13$ - to $16$ -year-olds.

$\bar{x}$ is an unbiased estimator of $μ$ if the average of all possible values of $\bar{x}$ is equal to $μ$ .

A study of the health of teenagers plans to measure the blood cholesterol levels of an SRS of $13$ - to $16$ -year-olds.

A larger sample gives more information about the distribution of values and thus it can be used to make more precise estimates.

More precise results also have less variability in the results.

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