A car manufacturer believes that marketing of eight-cylinder (V8) cars appeals mainly to men. A survey is conducted to ask, 'Do you prefer V8 cars?' The relative frequencies of men and women who answered 'yes' and 'no', respectively, are summarised in the following table:GenderoftherespondentPreferV8cars?

Men(yes)- 0.68 and (no)- 0.32 , Women(Yes) 0.56and No0.44

(a) There are two variables in this question, the gender of the respondents and their preference for V8 cars. What type of variables are they and what are their levels of measurement? Explain your answers. (b) Assume that 50 men and 50 women participated in the survey. At the 5% level of significance, is there a difference between males and females in terms of the proportion of people who prefer V8 cars? Specify the null and alternative hypotheses, calculate the observed test statistic first manually and then with R, make a statistical decision first based on the appropriate critical value and then on the p-value reported by R, and draw your conclusion. Are the conditions required by the test you performed likely satisfied? Explain your answer. (c) Repeat part (b), assuming this time that 200 men and 200 women participated in the survey. (d) Compare the details of your manual calculations in parts (b) and (c). Which details are the same, which are different? Discuss the effects of the sample size on your statistical decision. (e) Based on your confidence interval in part (c), does it seem reasonable to claim that the population proportion of men who prefer V8 cars is 10% higher than the corresponding population proportion of women? What about the claim that the population proportion of men who prefer V8 cars is 25% higher than the corresponding population proportion of women? Explain your answers.