Question: Testing for Categories with Different Proportions: Here are the observed frequencies from four categories: 5,10,10,20 .

Assume that we want to use a 0.05 significance level to test the claim that the four categories have proportions of 0.20,0.25,0.25 and $0.30 respectively.

1. What is the null hypothesis?
2. What re the expected frequencies for the four categories?
3. What is the value of the test statistic?
4. What is the critical value?
5. What do you conclude about the given claim?

8. Testing a Slot Machine. The author purchased a slot machine (Bally Model 809) and tested it by playing

it 1197 times. When testing the claim that the observed outcomes agree with the expected frequencies, a test statistic of \(\times 2=.185\) was obtained. There are 10 different categories of outcome, including a non-win, win jackpot, win with three bells and so on.

1. Find the critical value assuming that the significant level is 0.05
2. What can you conclude about the P-value from

6. No Smoking. The accompanying table summarizes successes and failures when subjects used different methods in trying to stop smoking. The determination of smoking or not smoking was made five months after the treatment was begun, and the data are based on results from the Centers for Disease Control and Prevention. Use the TI-8384 Plus results (see below) with a $0.05$ significance level to test the claim that the success is independent of the method used. If someone wants to stop smoking, does the choice of the method make a difference?

10. Solar Energy in Different Weather. A student of the author lives in a home with a solar electric system. At the same time each day, she collected voltage readings from a meter connected to the system and analysis of variance was used with readings obtained on three different types of day: sunny, cloudy, and rainy. The TI-83/8 Plus calculator results are below. Use a 0.05 significance level to test the claim that the mean voltage readings is the same for the three different types of day. Is there sufficient evidence to support a claim of different population means? We might expect that a solar system would provide more electrical energy on sunny days than on cloudy or rainy days. Can we conclude that sunny days results in greater amounts of electrical energy?

8. Interaction Effect. Test the null hypothesis that SAT scores are not affected by an interaction between gender and test (verbal/math). What do you conclude?

10. Effect of Type of SAT Test. Assume that SAT scores are not affected by an interaction between gender and the type of test (verbal/math). Is there sufficient evidence to support the claim that the type of test (verbal/math) has an effect on SAT scores?

8. Longevity. Refer to the accompanying table that lists the numbers of years that U.S. presidents and popes and British monarchs (since 1690) lived after their inauguration, election, or coronation. Determine whether the survival times for the three groups differ (Table based on data from Computer-Interactive Data Analysis by Lunn and McNeil, John Wiley & Sons.)

8. Testing a Slot Machine. The author purchased a slot machine (Bally Model 809), and tested it by playing it 1197 times. When testing the claim that the observed outcomes agree with the expected frequencies, a test statistic of \(\chi^2=9.185\) was obtained. There are 10 different categories of outcome, including no win, win jackpot, win with three bells and so on.

1. Find the critical value assuming that the significant level is 0.05
2. What can you conclude about the P-value from Table A-4 (see next sheet) if you know that the test statistic is \(\chi^2=8.185\) and there are 10 categories?
3. State a conclusion about the claim that the observed outcomes agree with the expected frequencies. Does the author's slot machine appears to be working correctly?

14. Which Treatment Is Better? A randomized controlled trial was designed to compare the effectiveness of splinting against surgery in the treatment of carpal tunnel syndrome. Results are given in the table below. The results are based on evaluations made one year after the treatment. Using a 0.01 significance level, test the claim that success is dependent of the type of treatment. What do the results suggest about treating carpal tunnel syndrome?

14. Archaeology. Skull Breadths from Different Epochs. Samples of head breadths were obtained by measuring skulls of Egyptian males from three different epochs, and the measurements are lsited below. Changes in head shape over time suggest that interbreeding occurred with immigrant populations. Use a $0.05$ significance level to test the claim that the different epochs do not all have the same mean. What do you conclude?

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