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Beginning in the second half of 2011 , there were widespread protests in many American cities that were primarily against Wall Street corruption and the gap between the rich and the poor in America. According to a Time Magazine/ABT SRBI poll conducted by telephone during October \(9-10,2011,86 \%\) of adults who were familiar with those protests agreed that Wall Street and lobbyists have too much influence in Washington (The New York Times, October 22, 2011). Assume that this percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 400 American adults who hold the opinion that Wall Street and lobbyists have too much influence in Washington. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

According to a poll, \(55 \%\) of Americans do not know that GOP stands for Grand Old Party (Time, October 17,2011 ). Assume that this percentage is true for the current population of Americans. Let \(\hat{p}\) be the proportion in a random sample of 900 Americans who do not know that GOP stands for Grand Old Party. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

Refer to Exercise 7.94. The current distribution of all monthly cable bills in the United States is approximately normal with a mean of $$\$ 82.69$$ and a standard deviation of $$\$ 11.17.$$ Find the probability that the average monthly cable bill for 23 U.S. households with cable is a. less than $$\$ 80$$ b. between $$\$ 75$$ and $$\$ 85$$ c. within $$\$ 5$$ of the population mean d. more than $$\$ 90$$

The Toyota Prius hybrid car is estimated to get an average of 50 miles per gallon (mpg) of gas. However, the gas mileage varies from car to car due to a variety of conditions, driving styles, and other factors and has been reported to be as high as \(70 \mathrm{mpg}\). Suppose that the distribution of miles per gallon for Toyota Prius hybrid cars has a mean of \(50 \mathrm{mpg}\) and a standard deviation of \(5.9 \mathrm{mpg}\). Find the probability that the average miles per gallon for 38 randomly selected Prius hybrid cars is a. more than \(51.5\) b. between 48 and 51 c. less than 53 d. greater than the population mean by \(2.5\) or more

In a large city, \(88 \%\) of the cases of car burglar alarms that go off are false. Let \(\hat{p}\) be the proportion of false alarms in a random sample of 80 cases of car burglar alarms that go off in this city. Calculate the mean and standard deviation of \(\hat{p}\), and describe the shape of its sampling distribution.