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Chapter 7: Problem 18

You need to select a simple random sample of two from six friends who will participate in a survey. Assume the friends are numbered \(1,2,3,4,5\), and 6 . Use technology to select your random sample. Indicate what numbers you obtained and how you interpreted them. If technology is not available, use the line from a random number table that corresponds to the day of the month on which you were born. For example, if you were born on the fifth day of any month, you would use line \(05 .\) Show the digits in the line and explain how you interpreted them.

Short Answer

Expert verified
The two friends selected for survey might be 3 and 5 by using technology or it might be 1 and 2 by the random number table method. Note that the selected friends might vary according to randomness.

Step by step solution

01

Understanding the Concept

A simple random sample is a subset of a statistical population, each member of which has an equal probability of being chosen. So, in simple random sampling, every combination of 2 friends amongst the 6 possible (which amounts to \(\binom{6}{2}\) or 15 possibilities) has an equal chance of getting selected.
02

Employing Technology

We can use a random number generator (like the ones available online or in scientific calculators) to obtain two distinct numbers between 1 and 6. Suppose we got 3 and 5. It means friend number 3 and friend number 5 are selected for the survey.
03

Implementing the Random Number Table Method

If technology is not available, another option is to use the random number table. Assume we were born on the 15th day of the month. On line 15, select the first two unique single-digit numbers encountered between 1 and 6. For instance, if line 15 reads 16245783... and so on, we pick 1 and 2. Thus, friend number 1 and friend number 2 are selected for the survey.

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Most popular questions from this chapter

According to a 2017 Pew Research Center report on voting issues, \(59 \%\) of Americans feel that the everything should be done to make it easy for every citizen to vote. Suppose a random sample of 200 Americans is selected. We are interested in finding the probability that the proportion of the sample who feel with way is greater than \(55 \%\). a. Without doing any calculations, determine whether this probability will be greater than \(50 \%\) or less than \(50 \%\). Explain your reasoning. b. Calculate the probability that the sample proportion is \(55 \%\) or more.

Has trust in the executive branch of government declined? A Gallup poll asked U.S. adults if they trusted the executive branch of government in 2008 and again in 2017 . The results are shown in the table. $$\begin{array}{|l|r|}\hline & \mathbf{2 0 0 8} & \mathbf{2 0 1 7} \\ \hline \text { Yes } & 623 & 460 \\\\\hline \text { No } & 399 & 562 \\\\\hline \text { Total } & 1022 & 1022 \\ \hline\end{array}$$ a. Find and compare the sample proportion for those who trusted the executive branch in 2008 and in 2017 . b. Find the \(95 \%\) confidence interval for the difference in the population proportions. Assume the conditions for using the confidence interval are met. Based on the interval, has there been a change in the proportion of U.S. adults who trust the executive branch? Explain.

In 2018 Gallup reported that \(52 \%\) of Americans are dissatisfied with the quality of the environment in the United States. This was based on a \(95 \%\) confidence interval with a margin of error of 4 percentage points. Assume the conditions for constructing the confidence interval are met. a. Report and interpret the confidence interval for the population proportion that are dissatisfied with the quality of the environment in the United States in 2018 . b. If the sample size were larger and the sample proportion stayed the same, would the resulting interval be wider or narrower than the one obtained in part a? c. If the confidence level were \(90 \%\) rather than \(95 \%\) and the sample proportion stayed the same, would the interval be wider or narrower than the one obtained in part a? d. In 2018 the population of the United States was roughly 327 million. If the population had been half that size, would this have changed any of the confidence intervals constructed in this problem? In other words, if the conditions for constructing a confidence interval are met, does the population size have any effect on the width of the interval?

The website www.mlb.com compiles statistics on all professional baseball players. For the 2017 season, statistics were recorded for all 663 players. Of this population, the mean batting average was \(0.236\) with a standard deviation of \(0.064\). Would it be appropriate to use this data to construct a \(95 \%\) confidence interval for the mean batting average of professional baseball players for the 2017 season? If so, construct the interval. If not, explain why it would be inappropriate to do so.

The mean weight of all professional NBA basketball players is \(218.8\) pounds. A sample of 50 professional basketball players has a mean weight of \(217.6\) pounds. Which number is \(\mu\), and which number is \(\bar{x}\) ?
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