Using R for Introductory Statistics : John Verzani

Significance tests 2158.5 On a number of highways a toll is collected for permission to travel on theroadway. To lessen the burden on drivers, electronic toll-collection systems are oftenused. An engineer wishes to check the validity of one such system. She arranges tosurvey a collection unit for single day, finding that of 5,760 transactions, the systemaccurately read 5,731. Perform a one-sided significance test to see if this is consistentwith a 99.9% accuracy rating at the 0.05 significance level. (Do you have any doubts thatthe normal approximation to the binomial distribution should apply here?)8.6 In Example 8.3 a count of 5,850 in the survey produced a p-value of 0.002363.What range of counts would have produced a p-value less than 0.05? (Start by askingwhat observed proportions in the survey would have such a p-value.)8.7 Historically, a car from a given company has a 10% chance of having a significantmechanical problem during its warranty period. A new model of the car is being sold. Ofthe first 25,000 sold, 2,700 have had an issue. Perform a test of significance to seewhether the proportion of these new cars that will have a problem is more than 10%.What is the p-value?8.8 A poll taken in 2003 of 200 Europeans found that only 16% favored the policies ofthe United States. Do a test of significance to see whether this is significantly differentfrom the 50% proportion of Americans in favor of these policies.8.2 Significance test for the mean (t-tests)Significance tests can also be constructed for the unknown mean of a parent population.The hypotheses take the formH 0 :µ=µ 0 , H A :µµ 0 , or µ≠µ 0 .For many populations, a useful test statistic isT takes the form of “observed” minus “expected,” divided by the standard error, wherethe expected value and the standard error are found under the null hypothesis.In the case of normally distributed initial data, the sampling distribution of T under thenull hypothesis is known to be the t-distribution with n−1 degrees of freedom. If n islarge enough, the sampling distribution of T is a standard normal by the central limittheorem. As both the t distribution and normal distribution are similar for large n, thefollowing applies to both assumptions.Test of significance for the population meanIf the data X 1 , X 2 , …, X n is an i.i.d. sequence from a Normal (µ, σ) distribution, or n islarge enough for the central limit theorem to apply, a test of significance forH 0 :µ=µ 0 , H A :µµ 0 , or µ≠µ 0can be performed with test statistic