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Chapter 9: Problem 77

For each of the following examples of tests of hypothesis about the population proportion, show the rejection and nonrejection regions on the graph of the sampling distribution of the sample proportion. a. A two-tailed test with \(\alpha=.10\) b. A left-tailed test with \(\alpha=.01\) c. A right-tailed test with \(\alpha=.05\)

Short Answer

Expert verified
The rejection regions for a two-tailed test with \(\alpha =0.10\) lie in the two extreme ends of the curve with 5% of data in each tail. For a left-tailed test with \(\alpha =0.01\), the leftmost 1% of the curve is the rejection region. For a right-tailed test with \(\alpha =0.05\), the extreme right 5% of the curve is the rejection region. The areas outside these regions in each case form the non-rejection region.

Step by step solution

01

Understand Rejection and Non-Rejection Regions

Rejection regions are the areas of a sampling distribution that would lead to rejecting the null hypothesis. They're determined by the level of significance \(\alpha\). Non-rejection regions, on the other hand, would lead to failing to reject the null-hypothesis.
02

Two-Tailed test

For a two-tailed test with \(\alpha =0.10\), the 10% is split between both tails of the distribution (5% in each tail). This means rejection regions lie in the two extreme ends of distribution curve, with 5% of data in each. The area in between these two tails forms the non-rejection region.
03

Left-Tailed test

In a left-tailed test with \(\alpha =0.01\), the whole \(\alpha\) lies in the left tail of the distribution. This means the leftmost 1% of distribution curve forms the rejection region and the area to the right of this comprises the non-rejection region.
04

Right-tailed Test

In a right-tailed test with \(\alpha =0.05\), the whole \(\alpha\) lies in the right tail of the distribution. The rightmost 5% of distribution curve forms the rejection region and the area to the left of this forms the non-rejection region.

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Most popular questions from this chapter

A company claims that its 8 -ounce low-fat yogurt cups contain, on average, at most 150 calories per cup. A consumer agency wanted to check whether or not this claim is true. A random sample of 10 such cups produced the following data on calories. 147 159 1 153 146 144 161 163 153 143 158 Test at the \(2.5 \%\) significance level whether the company's claim is true. Assume that the numbers of calories for such cups of yogurt produced by this company have an approximately normal distribution.

According to the American Diabetes Association (www.diabetes.org), \(23.1 \%\) of Americans aged 60 years or older had diabetes in 2007. A recent random sample of 200 Americans aged 60 years or older showed that 52 of them have diabetes. Using a \(5 \%\) significance level, perform a test of hypothesis to determine if the current percentage of Americans aged 60 years or older who have diabetes is higher than that in 2007 . Use both the \(p\) -value and the critical-value approaches.

According to the Magazine Publishers of America (www.magazine.org), the average visit at the magazines' Web sites during the fourth quarter of 2007 lasted \(4.145\) minutes. Forty-six randomly selected visits to magazine's Web sites during the fourth quarter of 2009 produced a sample mean visit of \(4.458\) minutes, with a standard deviation of \(1.14\) minutes. Using the \(10 \%\) significance level and the critical value approach, can you conclude that the length of an average visit to these Web sites during the fourth quarter of 2009 was longer than \(4.145\) minutes? Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.10\) ?

According to the University of Wisconsin Dairy Marketing and Risk Management Program (http:// future.aae.wisc.edu/index.html), the average retail price of 1 gallon of whole milk in the United States for April 2009 was \(\$ 3.084\). A recent random sample of 80 retailers in the United States produced an average milk price of \(\$ 3.022\) per gallon, with a standard deviation of \(\$ .274 .\) Do the data provide significant evidence at the \(1 \%\) level to conclude that the current average price of 1 gallon of milk in the United States is lower than the April 2009 average of \(\$ 3.084\) ?

A mail-order company claims that at least \(60 \%\) of all orders are mailed within 48 hours. From time to time the quality control department at the company checks if this promise is fulfilled. Recently the quality control department at this company took a sample of 400 orders and found that 208 of them were mailed within 48 hours of the placement of the orders. a. Testing at the \(1 \%\) significance level, can you conclude that the company's claim is true? b. What will your decision be in part a if the probability of making a Type I error is zero? Explain. c. Make the test of part a using the \(p\) -value approach and \(\alpha=.01\).
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