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Chapter 15: Problem 8

A Sample of Young Men. A government sample survey plans to measure the mean total cholesterol level of an SRS of men aged 20-34. The researchers will report the mean \(x\) from their sample as an estimate of the mean total cholesterol level \(\mu\) in this population. a. Explain to someone who knows no statistics what it means to say that \(x\) is an "unbiased" estimator of \(\mu\). b. The sample result \(x\) is an unbiased estimator of the population truth \(\mu\) no matter what size SRS the study uses. Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample.

Short Answer

Expert verified
An unbiased estimator means the sample mean equals the population mean on average. Larger samples reduce variability and increase trustworthiness in results.

Step by step solution

01

Defining an Unbiased Estimator

An unbiased estimator, like \( x \), means that over many repetitions or samples, \( x \) will, on average, equal the true population mean \( \mu \). In say, a hundred samples, if each sample's mean was calculated, the average of those means would be very close to \( \mu \). It's like aiming many arrows at a target; even if each arrow doesn't hit the bullseye, their average landing point should be around the center.
02

Impact of Larger Sample Size

Larger samples are more trustworthy because they tend to have less variability. With a larger sample, the calculated mean \( x \) will be closer to \( \mu \) more frequently. This is due to the Law of Large Numbers, where increasing the number of observations generally leads to results that are closer to the true population parameters.
03

Understanding Sampling Variability

Sampling variability refers to how much an estimator like \( x \) can change from one sample to another. A smaller sample typically has greater variability, meaning the mean \( x \) could vary widely from \( \mu \). Larger samples reduce this variability, making \( x \) consistently closer to \( \mu \), thus providing more reliable and confident estimates.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Random Sample (SRS)
To better understand what a Simple Random Sample (SRS) is, imagine putting everyone's name into a hat and picking out a certain number without looking. This process makes sure everyone has an equal chance of being chosen. That's basically how SRS works in statistics! Each member of the population has an equal opportunity to be part of the sample.
This approach is crucial because it helps avoid bias, making sure the sample truly represents the population. When a study relies on such a random method, it ensures the results are not skewed by favoritism towards one group or another, fortifying the representativeness of the findings.
Random sampling is the foundation of many statistical methods. By using an SRS, researchers can confidently draw inferences about a larger population, knowing their conclusions are derived from an unbiased selection of participants.
Law of Large Numbers
The Law of Large Numbers is an important concept in statistics. It tells us what happens as we take more and more samples. Imagine flipping a coin repeated times; initially, you might get a surprising string of heads or tails. However, as the number of flips increases, the results start averaging out to about 50% heads and 50% tails, which is expected.

In a similar way, when measuring something like cholesterol levels in a large group, the more individuals you measure, the closer your average result will likely be to the true average cholesterol level of the whole population. This happens because taking many measurements reduces the "noise" or random variations.
Consequently, the Law of Large Numbers reassures us that as we collect more data, our results become less erratic and more precise, stabilizing to reflect the true population values accurately.
Sampling Variability
Sampling Variability refers to the natural differences that arise when taking random samples from a population. Imagine you measure the average height of people in a city by picking a small group. Their average might be higher or lower than the city鈥檚 true average depending on who you randomly pick.

The smaller the group, the less representative it might be, which means more variability or fluctuation in the results. Therefore, if you measured another small group, its average could vary quite a bit from the first.
  • Larger samples tend to reflect more accuracy in reflecting the population.
  • Variability decreases with a bigger sample size, leading to more consistent results.
Ultimately, by reducing sampling variability with larger samples, researchers gain confidence that their estimates closely mirror the true population values, ensuring robust and reliable findings.

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Most popular questions from this chapter

Sampling Students. To estimate the mean score \(\mu\) of those who took the Medical College Admission Test on your campus, you will obtain the scores of an SRS of students. From published information, you know that the scores are approximately Normal, with standard deviation about \(10.6\). How large an SRS must you take to reduce the standard deviation of the sample mean score to 1 ?

Roulette. A roulette wheel has 38 slots, of which 18 are black, 18 are red, and two are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers is to choose red or black. A bet of \(\$ 1\) on red returns \(\$ 2\) if the ball lands in a red slot. Otherwise, the player loses the dollar. When gamblers bet on red or black, the two green slots result in losses. Because the probability of winning \(\$ 2\) is \(18 / 38\), the mean payoff from a \(\$ 1\) bet is twice \(18 / 38\), or \(94.7\) cents. Explain what the law of large numbers tells us about what will happen if a gambler makes very many bets on red.

Scores on the Evidence-Based Reading part of the SAT exam in a recent year were roughly Normal with mean 536 and standard deviation 102. You choose an SRS of 100 students and average their SAT Evidence-Based Reading scores. If you do this many times, the standard deviation of the average scores you get will be close to a. \(102 .\) b. \(102 / 100=1.02\). c. \(102 / \sqrt{100}=10.2\).

Annual returns on stocks vary a lot. The long-term mean return on stocks in the S\&P 500 is \(9.8 \%\), and the long-term standard deviation of returns is \(16.8 \%\). The law of large numbers says that a. you can get an average return higher than the mean \(9.8 \%\) by investing in a large number of the \(\mathrm{S} \& \mathrm{P}\) stocks. b. as you invest in more and more stocks chosen at random, your long-term average return on these stocks gets ever closer to \(9.8 \%\). c. if you invest in a large number of stocks chosen at random, your long-term average return will have approximately a Normal distribution.

Airline Passengers Get Heavier. In response to the increasing weight of airline passengers, the Federal Aviation Administration (FAA) in 2003 told airlines to assume that passengers average 195 pounds in the winter, including clothing and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 22 passengers. What is the approximate probability that the total weight of the passengers exceeds 4500 pounds? Use the four-step process to guide your work. (Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)
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