1) When the widget-making machine at WidgetCo is working properly, the widgets it produces are normally distributed with mean = 3 in. and standard deviation = .1 in. Every day, the quality-control inspector takes a simple random sample of 25 widgets. He will shut down the machine for an overhaul if the mean length of the widgets in the sample is not between 2.95 in. and 3.05 in.

a) Assuming the machine is working properly (i.e., = 3), what is the probability that the mean length of the widgets in the sample is between 2.95 and 3.05 in.?

b) What is the probability that the mean length of the widgets in the sample is not between 2.95 and 3.05 in.? (This is the probability that the machine will be shut down for an overhaul, even though it is running properly.)

c) Suppose the machine goes out of control and starts producing widgets with a mean length = 3.018 in. Repeat the calculation in a). (This result gives the probability that the machine will not be shut down for an overhaul, even though the widgets being produced are too long.)

2) A local power utility claims that the mean annual household electric bill for its subscribers is only $600. A consumer watchdog group wants to dispute this claim. Everyone agrees that the standard deviation of annual household electric bills is $150.

a) First the consumer group selects a simple random sample of 25 households and finds a mean annual electric bill of $648. Find a 95% confidence interval for the true mean annual electric bill, based on this sample.

b) From your result in a), is it reasonable to conclude (with 95% confidence) that the true mean annual bill for all subscribers must be over $600? Explain.

c) Some time later, a wealthy activist provides funding for a simple random sample of 250 households. The mean annual electric bill for this sample is $622. Find a 95% confidence interval for the true mean annual electric bill, based on this sample.

d) From your result in c), is it reasonable to conclude (with 95% confidence) that the true mean annual bill for all subscribers must be over $600? Explain.

e) Find the minimum sample size needed to estimate the true mean to within a $10 margin of error, with 95% confidence.

3) In an exit poll of 625 scientifically and randomly selected voters, 340 expressed support for Christine Carpetbagger for state senator.

a) Find the sample proportion of voters in the poll who support Carpetbagger.

b) Calculate a 98% confidence interval for the true proportion of all voters statewide who support Carpetbagger.

c) Based on this result, is it reasonable for Carpetbagger to declare victory? Explain.

4) Mileage figures (highway mpg) for 8 randomly selected 2007 Ford Predator SUV's are given below:

8.2 9.7 8.5 7.9 7.3 8.6 8.1 7.2

a) Find the sample mean and the sample standard deviation s.

b) Find a 90% confidence interval for the true mean mileage for all 2007 Predators