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A random sample of \(150\) recent donations at a certain blood bank reveals that \(82\) were type A blood. Does this suggest that the actual percentage of type A donations differs from \(40\% \), the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of \(.01\). Would your conclusion have been different if a significance level of \(.05\) had been used?

It is known that roughly \(2/3\) of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behaviour? The article 鈥淗uman Behaviour: Adult Persistence of Head-Turning Asymmetry鈥� (Nature, 2003: 771) reported that in a random sample of \(124\) kissing couples, both people in \(80\) of the couples tended to lean more to the right than to the left.

a. If \(2/3\) of all kissing couples exhibit this right-leaning behaviour, what is the probability that the number in a sample of \(124\) who do so differ from the expected value by at least as much as what was actually observed?

b. Does the result of the experiment suggest that the \(2/3\) figure is implausible for kissing behaviour? State and test the appropriate hypotheses.

With domestic sources of building supplies running low several years ago, roughly 60,000 homes were built with imported Chinese drywall. According to the article 鈥淩eport Links Chinese Drywall to Home Problems鈥� (New York Times, Nov. 24, 2009), federal investigators identified a strong association between chemicals in the drywall and electrical problems, and there is also strong evidence of respiratory difficulties due to the emission of hydrogen sulphide gas. An extensive examination of \(51\) homes found that \(41\) had such problems. Suppose these \(51\) were randomly sampled from the population of all homes having Chinese drywall.

a. Does the data provide strong evidence for concluding that more than \(50\% \) of all homes with Chinese drywall have electrical/environmental problems? Carry out a test of hypotheses using \(\alpha = .10\).

b. Calculate a lower confidence bound using a confidence level of \(99\% \) for the percentage of all such homes that have electrical/environmental problems.

c. If it is actually the case that \(80\% \) of all such homes have problems, how likely is it that the test of (a) would not conclude that more than \(50\% \) do?

A plan for an executive travelers鈥� club has been developed by an airline on the premise that \(5\% \) of its current customers would qualify for membership. A random sample of \(500\) customers yielded \(40\) who would qualify.

a. Using this data, test at level \(.01\) the null hypothesis that the company鈥檚 premise is correct against the alternative that it is not correct.

b. What is the probability that when the test of part (a) is used, the company鈥檚 premise will be judged correct when in fact \(10\% \) of all current customers qualify?

Pairs of P-values and significance levels, 伪, are given.

For each pair, state whether the observed P-value would lead to rejection of H₀ at the given significance level.

a.P颅-value = .084, 伪= .05

b.P颅-value = .003, 伪= .001

c.P-颅value = .498, 伪= .05

d.P-颅value = .084, 伪= .10

e.P-颅value = .039, 伪= .01

f.P-颅value = .218, 伪 = .10

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose the specifications state that mean strength of welds should exceed 100 lb/in² ; the inspection team decides to test H₀: 碌= 100 versus H_a: 碌> 100. Explain why it might be preferable to use

this Ha rather than 碌 < 100.

Each of a group of \(20\) intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let \(p\) denote the proportion of all such players who would prefer gut to nylon, and let \(X\) be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most \(50\% \) of all such players prefer gut. We simplify this to \({H_0}:p = .5\), planning to reject \({H_0}\) only if sample evidence strongly favors gut strings.

a. Is a significance level of exactly \(.05\) achievable? If not, what is the largest a smaller than \(.05\) that is achievable?

b. If \(60\% \) of all enthusiasts prefer gut, calculate the probability of a type II error using the significance level from part (a). Repeat if 80% of all enthusiasts prefer gut.

c. If \(13\) out of the \(20\) players prefer gut, should \({H_0}\) be rejected using the significance level of (a)?

A manufacturer of plumbing fixtures has developed a new type of washer less faucet. Let \(p = P\) (a randomly selected faucet of this type will develop a leak within \(2\) years under normal use). The manufacturer has decided to proceed with production unless it can be determined that \(p\) is too large; the borderline acceptable value of \(p\) is specified as \(.10\). The manufacturer decides to subject \(n\) of these faucets to accelerated testing (approximating \(2\) years of normal use). With \(X = \) the number among the \(n\) faucets that leak before the test concludes, production will commence unless the observed X is too large. It is decided that if \(p = .10\), the probability of not proceeding should be at most \(.10\), whereas if \(p = .30\) the probability of proceeding should be at most \(.10\). Can \(n = 10\) be used? \(n = 20\)? \(n = 25\)? What are the actual error probabilities for the chosen n?

Verify that the piecewise-defined function \(y = \left\{ {\begin{array}{*{20}{r}}{ - {x^2},}&{x < 0}\\{{x^2},}&{x \ge 0}\end{array}} \right.\) is a solution of the differential equation \(xy' - 2y = 0\) on \(( - \infty ,\infty )\).

In Problems \(11 - 14\) verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

\(y'' + y = tanx; y = - (cosx) ln(secx + tanx)\).