6.12 The mean is 5.4835 and the standard deviation is 0.1904. Table 4 in the textbook shows that with 23 - 1 = 22 degrees of freedom, the t value for a 99 percent confidence interval is 2.819. A 99 percent confidence interval includes the value 5.517 that is now accepted as the density of earth:

6.26 A 95 percent confidence interval is

6.28 For such a large sample (n = 10,000), we can use t* = 1.96 for the 95 percent confidence intervals:

6.44 Using a normal approximation to the binomial distribution with a 1.96 z value from Table 3 in the textbook, a 95 percent confidence interval just barely excludes 0.10:

If 10 percent of an infinite population of kegs are damaged, the probability that 12 or more kegs will be found damaged in a random sample of 50 kegs, using a normal approximation to the binomial distribution, is:

6.50 The sample proportion is x/n = 193/248 = 0.778. Using the sample proportions to estimate the standard deviation (and a 1.96 z value), a 95 percent confidence interval is

Because the sample is such a large part of the population, there is less chance for sampling error with a finite population in that it is less likely that the luck of the draw will give a sample with a disproportionate number of Republicans or Democrats; every Republican that is selected increases the chances of choosing a Democrat. (A finite population correction--which is beyond the scope of this textbook-- reduces the width of the confidence interval by 30 percent.)