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Chapter 7: Problem 14

Assume your class has 30 students and you want a random sample of 10 of them. A student suggests asking each student to flip a coin, and if the coin comes up heads, then he or she is in your sample. Explain why this is not a good method.

Short Answer

Expert verified
The coin flip method suggested is not a valid sample method because it does not guarantee the desired sample size of 10, it could result in a smaller or larger sample depending on the outcome of the flips. Also, this method does not constitute a truly random selection as each individual does not have an equal chance of being selected, thereby distorting the integrity and purpose of random sampling.

Step by step solution

01

Consider the Probability of Flipping a Coin

Let's consider the probability of flipping a coin. There's a 50% chance of getting heads and a 50% chance of getting tails. This means that, on average, out of 30 flips, about 15 heads might be expected, however this number is not fixed and can vary largely.
02

Consider the Sample Size

With this method, it's possible to get anywhere from 0 to 30 'heads', depending on how the coins fall. It's therefore not guaranteed to result in a sample size of exact 10. The sample size could be smaller or larger, thus failing to gather sufficient or precise data.
03

Consider the Concept of Random Sampling

Fundamentally, a random sample is designed to give every individual in a population an equal chance of being selected. In this case, if a student flips a tails, they are completely excluded from selection, while if they flip heads, they are automatically included. This does not constitute a truly random sample because the outcome on the first attempt determines inclusion or exclusion for the entire sample.

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Most popular questions from this chapter

An observation of the outcomes of rolling a die has about \(33.33 \%\) multiples of 3 and \(66.67 \%\) non-multiples of 3 , because two of the six outcomes are multiples of 3 ( 3 and 6 ) and four are not \((1,2,4\), and 5 ). a. Find the proportion of multiples of 3 in the following observations from a random roll of a die. Count carefully. $$ \begin{array}{llll} 31256 & 12351 & 34235 \\ 26346 & 43151 & 61322 \end{array} $$ b. Does the proportion found in part a represent \(\hat{p}\) (the sample proportion) or \(p\) (the population proportion)? c. Find the error in this estimate, the difference between \(\hat{p}\) and \(p\) (or \(\hat{p}-p\) ).

Explain the difference between sampling with replacement and sampling without replacement. Suppose you have a deck of 52 cards and want to select two cards. Describe both procedures.

According to an article in randomhistory.com, only \(18 \%\) of all millionaires in the world have a master's degree. Suppose a conclave of millionaires contains 150 millionaires that were randomly sampled from the population of millionaires. Use the Central Limit Theorem (and the Empirical Rule) to find the approximate probability that the conclave will have a proportion of millionaires with master's degrees that is more than two standard errors below \(0.18\). You can use the Central Limit Theorem because the millionaires were randomly sampled; the population is more than 10 times 150; and \(n\) times \(p\) is 27 and \(n\) times \((1\) minus \(p\) ) is 123 , and both are more than 10 .

A survey was conducted to ask whether tax benefits for senior citizens should be continued or stopped. Only clubs were visited to collect data. Do you think this would introduce bias? Explain.

In November 2011 , a Pew Poll showed that 1241 out of 2001 randomly polled people in the United States favor the death penalty for those convicted of murder. Assuming the conditions for using the CLT were met, answer these questions. $$ \begin{array}{|lrrlc|} \hline \text { Sample } & \mathrm{X} & \mathrm{N} & \text { Sample } \mathrm{p} & 958 \mathrm{CI} \\ 1 & 1241 & 2001 & 0.620190 & (0.598925,0.641455) \\ \hline \end{array} $$ Minitab Output a. Using the Minitab output given, write out the following sentence, filling in the blanks. I am \(95 \%\) confident that the population proportion favoring the death penalty is between \(\longrightarrow\) and . Report each number correct to three decimal places. b. Is it plausible to claim that a majority favor the death penalty? Explain.
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