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Chapter 7: Problem 103

If you walked around your school campus and asked people you met how many keys they were carrying, would you be obtaining a random sample? Explain.

Short Answer

Expert verified
No, by asking people met around the school campus how many keys they are carrying would not be obtaining a random sample. This is due to the fact that not everyone in the population (i.e. everyone who goes to the school) has an equal chance of being selected, a condition necessary for the sample to be considered random.

Step by step solution

01

Understanding the situation

There is a collection of data involved in the exercise, which involves asking people met around a school campus about the number of keys they are carrying.
02

Understanding the concept of a random sample

In a random sample, every individual in the population has an equal chance to be selected. The best way to achieve a random sample is by having a list of everyone in the population and using a random method to select individuals.
03

Analysis of the given situation

In the given exercise, while asking people met around a school campus about the number of keys they carry, there are likely biases affecting the sampling. For instance, only people who are present at the campus during the time the question is asked and are willing to respond to the question are included, not giving an equal chance to everyone in the population to be selected.

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Most popular questions from this chapter

According to a 2018 Rasmussen Poll, \(40 \%\) of American adults were very likely to watch some of the Winter Olympic coverage on television. The survey polled 1000 American adults and had a margin of error of plus or minus 3 percentage points with a \(95 \%\) level of confidence. a. State the survey results in confidence interval form and interpret the interval. b. If the Rasmussen Poll was to conduct 100 such surveys of 1000 American adults, how many of them would result in confidence intervals that included the true population proportion? c. Suppose a student wrote this interpretation of the confidence interval: "We are \(95 \%\) confident that the sample proportion is between \(37 \%\) and \(43 \%\)." What, if anything, is incorrect in this interpretation?

Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$\begin{array}{|l|l|l|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\\\\hline & 936 & 607 \\ \hline\end{array}$$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$\begin{array}{|l|c|c|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\\\\hline & 94 & 61 \\ \hline\end{array}$$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.

From Formula \(7.2\), an estimate for margin of error for a \(95 \%\) confidence interval is \(m=2 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) where \(\mathrm{n}\) is the required sample size and \(\hat{p}\) is the sample proportion. Since we do not know a value for \(\hat{p}\), we use a conservative estimate of \(0.50\) for \(\hat{p}\). Replace \(\hat{p}\) with \(0.50\) in the formula and simplify.

Suppose it is known that \(20 \%\) of students at a certain college participate in a textbook recycling program each semester. a. If a random sample of 50 students is selected, do we expect that exactly \(20 \%\) of the sample participates in the textbook recycling program? Why or why not? b. Suppose we take a sample of 500 students and find the sample proportion participating in the recycling program. Which sample proportion do you think is more likely to be closer to \(20 \%\) : the proportion from a sample size of 50 or the proportion from a sample size of \(500 ?\) Explain your reasoning.

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.
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