A well known manufacturer of flat screen, HDTV screens is considering changing the width to height ratio on its product. In particular they want to see if the ratio of the dimension of its screens is of importance to the consumer. The marketing department had several different versions of their screens designed with different width to height ratios. Their traditional screen has a ratio of 4:3. The design changes to increase or decrease the ratio involves nothing more Than adding or deleting pixels and rearranging some of them to meet the ratio dimensions, the size of the body that holds the screen would remain essentially unchanged.

The marketing department collected data from a sample of 100 potential customers and ran several tests on the data to help them make suggestions to senior management on a possible change to the screen design. The marketing department had 11 different screen designs made with different ratios and asked the potential customers to respond with some identifying information, and, which design they thought was the most appealing. The results of the sample were categorized in several different ways for their analyses

Problem 1: First, the marketing department was not even sure if there would be a preference for different screen ratios. The sample data was categorized into 5 different ratio groups and if there were no preference they expected to find 20% of the potential consumers selecting each of the categories. The following is the data from the sample of 10Qpotential consumers. At the 0.01 level of significance test to see if the results met their "no preference" expectations or if there was actually a preference for one or more of the screen ratio options. The value of the test statistic for this test was calculated and is given as 22.60

Problem 2: Subsequently the marketing department was not sure if the screen ratios were independent or dependent of the age groups of the people who would find the screen design most appealing. The following set of data was collected to test for independence to see if the screen ratio was independent of age groups. Conduct this test for independence with an \(\alpha \) of 0.10. The value of the test statistic for this test-was calculated and is given as 3.69.

Problem 3: Continuing on, the marketing department wanted to see if there was a difference in the proportion of males and females in their preferences for the screen ratios of the TVs. 66 of the 100 potential consumers who-participated in the sample were female, 34 were male. When asked the general question: "Do you prefer a TV with a screen ratio design that is significantly larger than 4:3?" 44 of the females responded yes, and 28 of the men responded yes. Test to see if there is a significant difference in the proportions of females and males who responded positively to this question at the 0.05 level of significance.

Problem 4: The marketing department has historical information that says that 70% of all people surveyed would respond positively to the question "Do you prefer a TV with a screen ratio design that is significantly greater than 4:3?" As we can see from the last question, this time there was a total of 72 of the 100 people who responded positively to this question. At the 0.05 level of significance has there been a significant increase in the proportion of people who would now respond positively to the same question?

Problem 5: The marketing department also has historical information that indicates that the preferred screen ratios follow a normal distribution with a mean of 1.62:1 and a standard deviation of 0.08.

1. If a single person was randomly selected (not involved in the sample survey of 100, but from the general population), what is the probability that it would show that a screen ratio of greater than 4:3 was preferred?
2. If a sample of 5 were randomly selected (not involved in the sample survey of 100, but from the general population), what is the probability that the sample mean would show that a screen ratio of 4:3 or greater was preferred?

Problem 6: Returning to the information in question 5 above, if 70% of all people would normally respond positively to the question "Do you prefer a TV screen design that has a ratio significantly greater than 4:3?" What is the probability that at least 2 out of 8 randomly selected people (not involved in the sample survey of 100, but from the general population) would say yes?

Problem 7: Returning to the contingency table shown in problem 2 above, if one of the sample responses was randomly selected from the group of 100 responses, determine the following probabilities:

1. What is the probability that the randomly selected person was someone who preferred the 1:1 - 1.3:1 screen ratio given that they were in the teenage age group?
2. What is the probability that the randomly selected person was a young adult and preferred the 1.4:1 - 1.6:1 screen ratio?
3. What is the probability that the randomly selected person was someone who was middle aged or preferred the 1.4:1 - 1.6:1 screen ratio?
4. What is the probability that the randomly selected person was a senior?

Problem 8: In the Middle aged group (36-55 yr old) who preferred the 1.7:1 – 2:1 screen ratios, there were 4 men and 7 women. The following were their preferences:

From this data test to see if the mean preferred screen ratio of the women in this group is greater than the preferred screen ratio of the men at the 0.05 level of significance. The degrees of freedom for this question have been calculated and are given to be 8. (Do not use any historical data from any of the previous problems, use only this sample data.)

Solution: We have the following table:

Problem 9: The marketing department wants to analyze the group that preferred the 1:1 - 1.3:1 screen ratios. The following data show the actual screen ratio preferences by age groups. The critical value for this test was calculated and is given as 6.15.

With an \(\alpha \) of 0.01 is there a difference between any of the mean screen ratios within these groups?

Problem 10: Finally, the marketing department selected 8 of the individuals for a special test concerning some proposed commercial advertisements for television viewing concerning new ratio designs. Originally they had all selected screen ratios of 1.6:1 or less. Then they were shown 5 different 30-second television commercials that emphasized images with designs that used screen ratios of 1.6:1 or greater. Following the viewing of the commercials they were then asked to select their preferred screen ratios again and their preferred screen ratios were recorded. The following is the data from this test.

With a confidence level of 95% ( \(\alpha \) = 0.05) test to see if viewing the proposed television commercials made a difference in the preferred screen ratios of this sample of 8 potential consumers.

Problem 11: Write a couple of summary paragraphs that you might present to senior management summarizing the results of these analyses and any recommendations you might have about the possible changes to the dimensions or ratios of their flat screen HDTV’s based on your collective analysis.

Problem 12: In your own words (please do not give me a quote from the textbook) write a paragraph defining "Inferential Statistics".

Problem 13: In your own words (please do not give me a quote from the textbook) write a paragraph defining and explaining the use of the p-value"

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