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# Business Statistics - Null and Alternative Hypothesis

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Writing Hypotheses

Read the situation. Then write the hypotheses in correct mathematical notation. Do not conduct any statistical tests. Just write the hypotheses. Insert your answers between the problems.

Here are some things to keep in mind:

1) On the Hypothesis Testing Worksheet, all you need to do is write the null and alternative hypotheses for each situation.

2) The null hypothesis will always be "=".

3) You can use either "≠" or "not =" for "does not equal". Greater than and less than is ">" or "<", respectively.

4) The alternative hypothesis wil be "not =" (2 tailed test) of ">" or "<" (one tailed test).

5) When determining what the null and alternative hypotheses are, realize that the alternative is the new information, what you are trying to prove. The null is what has been believed to be true up until now.

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1) A bowler who has averaged 196 pins in the past year is asked to experiment with a ball made of a new kind of material. He rolls several games with the new ball. Has the new ball improved his game?

2) An advertisement claims that chewing NoCav gum reduces cavities. To test the claim, you conduct a study in which participants who chew the gum are compared to the national average of 3 cavities found per year.

3) In a speech to the Chamber of Commerce, a city councilman claims that in his city less than 15% of the adult male population are unemployed. An opponent in the upcoming election wants to test the councilman's claim.

4) The councilman is starting to get worried about the upcoming election. He has enjoyed 63% support for several years, but the political climate has been changing. He wants to know if his support has changed.

5) A production process is considered to be under control if the machine parts it makes have a mean length of 35.50 mm with a standard deviation of 0.45 mm. Whether or not the process is under control is decided each morning by a quality control engineer who bases his decision on a random sample of size 36. Should he ask for an adjustment of the machine on a day when he obtains a mean of 35.62 mm?

6) Jim, the owner of Jim's Grocery, knows that Plain Chips have always outsold Spicy chips. However, sales of Spicy chips have been increasing. Jim wants to determine if the average weekly sales of Spicy chips have indeed surpassed that of Plain chips.

7) Jim now wants to know if Plain and Spicy chips have the same percentage of defective product (i.e. underfilled bags, torn bags, wrong flavor in the bags, etc.).

8) The Great Vehicle Co. just introduced New SUV, claiming it can pull more weight than Old SUV. After testing 150 vehicles of each model, Old SUV had a mean pull weight of 5032 pounds with a standard deviation of 72 pounds. New SUV had a mean pull weight of 5462 pounds with a standard deviation of 154 pounds. Is the claim valid at a .05 level of significance?

9) The Great Vehicle Co. has a competitor, Amazing Autos, that claims people who purchase its competing vehicle, the Sport Off Road Vehicle (SORV), have higher customer satisfaction than New SUV. Out of 736 people who purchased the SORV last month, 534 said they were satisfied. Out of 521 people who purchased New SUV last month, 375 said they were satisfied. Is there a higher percentage of people who are satisfied with the SORV than with New SUV?

10) The Great Vehicle Company wants to counter Amazing Autos's claim by making its own claim that New SUV has a lower percentage of defective vehicles. The research team tested 536 vehicles of each model and found that SORV had 53 defective units, while New SUV had only 51 defective units.

Statistics Problems

1) Ask 10 people (get 5 males and 5 females) the following questions

A) Their ages

B) How many vitamins they take daily

C) How many carbonated sodas they drink each day

D) How many alcoholic beverages they drink per month

E) Write your own question. Ask your participants if they agree with something or if they do something. For example, you may want to ask them if they eat popcorn when they go to the movies or if they support a political issue. It must be a yes/no question.

SHOW & SAVE YOUR DATA - You will use the data you gathered above for the problems below

Practice Problems:

1) Use your data from above. This week assume that historically the average person takes 3 vitamins on a daily basis. Conduct a hypothesis test analysis to determine if 3 is still the correct average number. Write your hypotheses in correct statistical notation. Finally use the important numbers from your output to explain your results. Use alpha = 0.05. Post only the relevant numbers, not all of the output; then explain your results.

2) Use your data from above. Analyze if more than 58% support an issue or partake in an activity. (Question E above). Write the hypotheses. Show the relevant numbers. Then explain your results. Use alpha = 0.05.

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#### Solution Summary

This solution is comprised of a detailed explanation of defining the null and alternative hypotheses for one sample t test, one sample z test, two sample t test and z test for testing the two proportion. The hypothetical data was obtained to testing two hypothesis (one based on t test for mean and one based on z test for proportion). A detailed explanation if provided for all questions.

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## Distribution Analysis: Statistics for Business Textbook Problems

6.8 For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.
a. z>1.96
b. z>1.645
c. z>2.575
d. z<-1.28
e. z<-1.645 or z>1.645
f. z<-2575 or z>2.575
g. For each of the rejection regions specified in parts a-f, what is the probability that a Type I error will be made?

6.10 Play Golf America Program. The Professional Golf Association (PGA) and Golf Digest have developed the Play Golf America program, in which teaching professionals at participating golf clubs provide a free 10-minute lesson to new customers. According to Golf Digest (July 2008), golf facilities that participate in the program gain, on average, \$2,400 in green fees, lessons, or equipment expenditures. A teaching professional at a golf club believes that the average gain in greens fees, lessons, or equipment expenditures for participating golf facilities exceeds \$2,400.
a. In order to support the claim made by the teaching professional, what null and alternative hypotheses should you test?
b. Suppose you select alpha = 0.05. Interpret this value in the words of the problem.
c. For alpha = 0.05, specify the rejection region of a large sample test.

6.20 A random sample of 100 observations from a population with standard deviation 60 yielded a sample mean of 110.
a. Test the null hypothesis that u = 100 against the alternative hypothesis that u > 100 using alpha = 0.05. Interpret the results of the test.
b. Test the null hypothesis that u = 100 against the alternative hypothesis that u =/ 100 using alpha = 0.05. Interpret the results of the test.
c. Compare the results of the two tests you conducted. Explain why the results differ.

6.22 Accounting and Machiavellianism. Refer to the Behavioral Research in Accounting (Jan. 2008) study of Machiavellian traits in accountants, Exercise 5.17 (p. 279). A Mach rating score was determined for each in a random sample of 122 purchasing managers with the following results: x-bar = 996, s = 12.6. Recall that a director of purchasing at a major firm claims that the true mean Mach rating score of all purchasing managers is 85.
a. Suppose ou want to test the director's claim. Specify the null and alternative hypothesis for the test.
b. Give the rejection region for the test using alpha = 0.10.
c. Find the value of the test statistic.
d. Use the result, part c, to make the appropriate conclusion.

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