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Chapter 7: Problem 47

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.

Short Answer

Expert verified
a. The probability will be more than 50% since 55% is less than the given proportion of 59%. \nb. The probability that the sample proportion is 55% or more can be calculated by first finding the Z-score and then checking a standard normal table to find the exact probability.

Step by step solution

01

Analyze and make a preliminary assessment

Looking at the overall component of the exercise, a rough estimate without calculations can provide a simple understanding. As per the data, since 59% American believe in easy voting rights and we're looking for the probability of the sample proportion being greater than 55%, therefore, the probability will be higher than 50% as 55% is less than 59%.
02

Establish the conditions

In this situation, the sample estimate will follow a Normal distribution. The population proportion is \(p = 0.59\). Hence, we know the following: \n - sample size \(n = 200\)\n - number of successes \(X = n*prop = 200*0.55 = 110\)\n - The mean of the sample distribution \( \mu = p = 0.59\)\n - The standard deviation \( \sigma = \sqrt{p*(1-p)/n} = \sqrt{0.59*0.41/200}\)
03

Calculate the standard normal variable

To find the probability that the sample proportion is more than 0.55, use the following formula for Z-score:\n \(Z = (X - \mu)/\sigma \) where X = sample proportion = 0.55\nSubstituting the values gives us the Z-score
04

Find Probability

Utilize standard normal tables or a suitable software, the probability corresponding to this Z-score is calculated, which will yield the chance that the proportion of the sample who feel this way is greater than 55%.

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Most popular questions from this chapter

In 2003 and 2017 Gallup asked Democratic voters about their views on the FBI. In \(2003,44 \%\) thought the \(\mathrm{FBI}\) did a good or excellent job. In \(2017,69 \%\) of Democratic voters felt this way. Assume these percentages are based on samples of 1200 Democratic voters. a. Can we conclude, on the basis of these two percentages alone, that the proportion of Democratic voters who think the FBI is doing a good or excellent job has increase from 2003 to \(2017 ?\) Why or why not? b. Check that the conditions for using a two-proportion confidence interval hold. You can assume that the sample is a random sample. c. Construct a \(95 \%\) confidence interval for the difference in the proportions of Democratic voters who believe the FBI is doing a good or excellent job, \(p_{1}-p_{2}\). Let \(p_{1}\) be the proportion of Democratic voters who felt this way in 2003 and \(p_{2}\) be the proportion of Democratic voters who felt this way in 2017 . d. Interpret the interval you constructed in part c. Has the proportion of Democratic voters who feel this way increased? Explain.

Ignaz Semmelweiss (1818-1865) was the doctor who first encouraged other doctors to wash their hands with disinfectant before touching patients. Before the new procedure was established, the rate of infection at Dr. Semmelweiss's hospital was about \(10 \%\). Afterward the rate dropped to about \(1 \%\). Assuming the population proportion of infections was \(10 \%\), find the probability that the sample proportion will be \(1 \%\) or less, assuming a sample size of \(200 .\) Start by checking the conditions required for the Central Limit Theorem to apply.

The website scholarshipstats.com collected data on all 5341 NCAA basketball players for the 2017 season and found a mean height of 77 inches. Is the number 77 a parameter or a statistic? Also identify the population and explain your choice.

Just the Boys Refer to Exercise \(7.77\) for information. This data set records results just for the boys. $$\begin{array}{|lcc|}\hline & \text { Preschool } & \text { No Preschool } \\\\\hline \text { Grad HS } & 16 & 21 \\\\\hline \text { No Grad HS } & 16 & 18 \\\\\hline\end{array}$$ a. Find and compare the percentages that graduated for each group, descriptively. Does this suggest that preschool was linked with a higher graduation rate? b. Verify that the conditions for a two-proportion confidence interval are satisfied. c. Indicate which one of the following statements is correct. i. The interval does not capture 0 , suggesting that it is plausible that the proportions are the same. ii. The interval does not capture 0 , suggesting that it is not plausible that the proportions are the same. iii. The interval captures 0 , suggesting that it is plausible that the population proportions are the same. iv. The interval captures 0 , suggesting that it is not plausible that the population proportions are the same. d. Would a \(99 \%\) confidence interval be wider or narrower?

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.
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