LSuppose that insurance companies did a survey. They randomly surveyed 400 drivers and found that 340 claimed they always buckle up. We are interested in the population proportion of drivers who claim hey always buckle up. NOTE: If you are using a Student's tdistribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) B Part (a) (i) Enter an exact number as an integer, fraction, or decimal. x = x (ii) Enter an exact number as an integer, fraction, or decimal. '7: X (iii) Round your answer to four decimal plaoes, p' = X (rounded to four decimal places) El Part (b) In words, dene the random variables X and P'. o X is the number of people who claim they buckle up, and P' is the proportion of people in the sample who buckle up. \" X is the proportion of people in the sample who do not buckle up, and P' is the number of people who do not buckle up. "' X is the proportion of people in the sample who claim they buckle up, and P' is the number of people who buckle up. '" X is the number of people who do not buckle up, and P' is the proportion of people in the sample who do not buckle up. u! Correctl The random variable of interest, X, is the number of people who buckle up, and P'represents the proportion of the sample who buckle up. El Part (c) Which distribution should you use for this problem? (Round your answer to four decimal places.) P'~ x( x. x) Explain your choice. o The binomial distribution should be used because there are two outcomes, buckle up or do not buckle up. A The Students tdistribution should be used because \\/ npq S 10, which implies a small sample. The Students tdistribution should be used because we do not know the standard deviation. i: run \"5, Which distribution should you use for this problem? (Round your answer to four decimal places.) P'~ x( x. x) Explain your choice. o The binomial distribution should be used because there are two outcomes, buckle up or do not buckle up. The Student's tdistribution should be used because \\/ npq s 10, which implies a small sample. ' ' The Student's t-distribution should be used because we do not know the standard deviation. \" The normal distribution should be used because we are interested in proportions and the sample size is large. X Incorrect. While this may represent a binomial exercise, we use the standard normal distribution to approximate binomial condence intervals. El em (a) Construct a 95% condence interval for the population proportion who claim they always buckle up. (i) State the condence interval. (Round your answers to four decimal places.) ( x . x ) (ii) Sketch the graph. 3 = X C_L_ = x 2 \"'9 II 3: (iii) Calculate the error bound. (Round your answer to four decimal places.) Part (d) Construct a 95% confidence interval for the population proportion who claim they always buckle up. (i) State the confidence interval. (Round your answers to four decimal places.) X x ) (ii) Sketch the graph. = X C.L. = X = X X X P (iii) Calculate the error bound. (Round your answer to four decimal places.) X - Part (e) If this survey were done by telephone, list three difficulties the companies might have in obtaining random results. Children who do not answer the phone will not be included. Individuals may choose to participate or not participate in the phone survey. The individuals in the sample may not accurately reflect the population. Only people over the age of 24 would be included in the survey. O Individuals may not tell the truth about buckling up. X