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In a batch chemical process, two catalysts arc being compared for their effect on the output of the process reaction. A sample of \(\lfloor 2\) batches was prepared using catalyst 1 and a sample of 10 batches was obtained using catalyst \(2 .\) The 12 batches for which catalyst 1 was used gave an average yield of 85 with a sample standard deviation of \(4,\) and the second sample gave an average of 81 and a sample standard deviation of \(5 .\) Find a \(90 \%\) confidence interval for the difference between the population means, assuming that the: populations art: approximately normally distributed with equal variances.

Students may choose between a 3-semester-hour course in physics without labs and a 4-semester-hour course with labs. The final written examination is the same for each section. If 12 students in the section with labs made an average examination grade of 84 with a standard deviation of \(4,\) and 18 students in the section without labs made an average: grade of 77 with a standard deviation of \(6,\) find a \(99 \%\). confidence interval for the difference between the average grades for the two courses. Assume the populations to be approximately normally distributed with equal variances.

(a) A random sample of 200 voters is selected and 114 are found to support an annexation suit. Find the \(96 \%\) confidence interval for the fraction of the voting population favoring the suit. (b) What can we assert with \(96 \%\) confidence about the possible size of our error if we estimate the fraction of voters favoring the annexation suit to be \(0.57 ?\)

A new rocket-launching system is being considered for deployment of small, short-range rockets. The existing system has \(p=0.8\) as the probability of a successful launch. A sample of 40 experimental launches is made with the new system and 34 are successful. (a) Construct a \(95 \%\) confidence interval for \(p\). (b) Would you conclude that the new system is better?

A study is to be made to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant. How large a sample is needed if one wishes to be at least \(95 \%\) confident that the estimate is within 0.04 of the true proportion of residents in this city and its suburbs that favor the construction of the nuclear power plant?

Ten engineering schools in the United States were surveyed. The sample contained 250 electrical engineers, 80 being women; 175 chemical engineers, 40 being women. Compute a \(90 \%\) confidence interval for the difference between the proportion of women in these two fields of engineering. Is there a significant difference between the two proportions?

According to USA Today (March 17. \(\lfloor 997\) ), women made up \(33.7 \%\) of the editorial staff at local TV stations in 1990 and \(36.2 \%\) in \(1994 .\) Assume 20 new employees were: hired as editorial staff. (a) Estimate the number that would have been women in each year, respectively, (b) Compute a \(95 \%\) confidence interval to see if there is evidence that the proportion of women hired as editorial staff in 1994 was higher than the proportion hired in 1990 .

A manufacturer of car batteries claims that his batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes of 1.9 \(2.4,3.0,3.5,\) and 4.2 years, construct a \(95 \%\) confidence interval for \(\sigma^{2}\) and decide if the manufacturer's claim that \(\sigma^{2}=1\) is valid. Assume the population of battery lives to be approximately normally distributed.

A random sample of 20 students obtained a mean of \(\bar{x}=72\) and a variance of \(s \sim^{2}=16\) on a college placement test in mathematics. Assuming the scores to be normally distributed, construct a \(98 \%\) confidence interval for \(\sigma^{2}\).

Suppose that there are \(n\) trials \(x_{1}, x_{2}, x_{n}\) from a Bernoulli process with parameter \(p\), the probability of a success. That is, the probability of \(r\) successes is given by \(\left(\begin{array}{l}n \\\ r\end{array}\right) p^{r}(1-p)^{n-F} .\) Work out the maximum likelihood estimator for the parameter \(p\).