2. Decide whether the following statements are true or false. Explain your reasoning. a) For a given standard error, lower confidence levels produce wider confidence intervals. b) If you increase sample size, the width of confidence intervals will increase. c) The statement, "the 95% confidence interval for the population mean is (350, 400)", is equivalent to the statement, "there is a 95% probability that the population mean is between 350 and 400". d) To reduce the width of a confidence interval by a factor of two (i.e., in half), you have to quadruple the sample size. e) Assuming the central limit theorem applies, confidence intervals are always valid. f) The statement, "the 95% confidence interval for the population mean is (350, 400)" means that 95% of the population values are between 350 and 400. g) If you take large random samples over and over again from the same population, and make 95% confidence intervals for the population average, about 95% of the intervals should contain the population average. h) If you take large random samples over and over again from the same population, and make 95% confidence intervals for the population average, about 95% of the intervals should contain the sample average. i) It is necessary that the distribution of the variable of interest follows a normal curve. j) A 95% confidence interval obtained from a random sample of 1000 people has a better chance of containing the population percentage than a 95% confidence interval obtained from a random sample of 500 people. k) If you go through life making 99% confidence intervals for all sorts of population means, about 1% of the time the intervals won't cover their respective population means.