Problem 7.20: Suppose that events A and B are mutually exclusive with \(\Pr \left( A \right)=\frac{1}{2}\) and \(\Pr \left( B \right)=\frac{1}{3}\).

1. Are A and B independent?
2. Are A and B complementary?

Problem 7.34: Two fair coins are tossed. Define

1. Find \(\Pr \left( A \right)\)
2. Find \(\Pr \left( B \right)\)
3. Compute \(\Pr \left( A\cap B \right)\)
4. Find \(\Pr \left( A\cup B \right)\)

Problem 7.36: In a recent election, 55% of the voters were republican, and 45% were not. Of the republicans, 80% voted for candidate X, and of the non-republican, 10% voted for candidate X. Consider a randomly selected voter. Define

1. Write values for \(\Pr \left( A \right),\Pr \left( {{A}^{C}} \right),\Pr \left( B|A \right)\) and \(\Pr \left( B|{{A}^{C}} \right)\).
2. Find \(\Pr \left( A\cap B \right)\) and write in word what outcome it represents.
3. Find \(\Pr \left( {{A}^{C}}\cap B \right)\) and write in word what outcome it represents.
4. Find \(\Pr \left( B \right)\)
5. What percentage of the vote did the Candidate X receive?

Problem 8.51: Weights X of men in a certain age group have a normal distribution with mean \(\mu =180\) pounds and a standard deviation of \(\sigma =20\) pounds. Find each of the following probabilities:

1. \(\Pr \left( X\le 200 \right)=\Pr \left( \frac{X-180}{20}\le \frac{200-180}{20} \right)=\Pr \left( Z\le 1 \right)=0.841345\)
   where Z has a standard normal distribution.
2. \(\Pr \left( X\le 165 \right)=\Pr \left( \frac{X-180}{20}\le \frac{165-180}{20} \right)=\Pr \left( Z\le -0.75 \right)=0.226627\)
3. \(\Pr \left( X>165 \right)=1-\Pr \left( X\le 165 \right)=1-0.226627=0.773373\)

Problem 8.58: Find the following probabilities for Verbal SAT test scores X , for which the mean is 500 and the standard deviation is 100. Assume SAT scores are described by a normal curve

1. \(\Pr \left( X\le 500 \right)=\Pr \left( \frac{X-500}{100}\le \frac{500-500}{100} \right)=\Pr \left( Z\le 0 \right)=0.5\)
2. \(\Pr \left( X\le 650 \right)=\Pr \left( \frac{X-500}{100}\le \frac{650-500}{100} \right)=\Pr \left( Z\le 1.5 \right)=0.933193\)
3. \(\Pr \left( X\ge 700 \right)=\Pr \left( \frac{X-500}{100}\ge \frac{700-500}{100} \right)=\Pr \left( Z\ge 2 \right)=1-\Pr \left( Z<2 \right)=0.02275\)
4. \(\Pr \left( 500\le X\le 700 \right)=1-\Pr \left( X\le 500 \right)-\Pr \left( X\ge 700 \right)=0.044057\)

Problem 9.13: A medical researcher wants to estimate the difference in the proportions of women with high blood pressure for women who use oral contraceptives versus women who don’t use oral contraceptives. In a observational study involving a sample of 900 women the researcher finds that 15% of the 500 women who used oral contraceptives had high blood pressure, whereas only 10% of the 400 women who didn’t use oral contraceptive had high blood pressure.

1. What is the research question?
2. What is the population parameter in this study?
3. What is the value of the sample estimate in this study?

Problem 9.66: Explain which parameter is being described in each of the following situations: \(\mu ,{{\mu }_{D}}\) or \({{\mu }_{1}}-{{\mu }_{2}}\)

1. \({{\mu }_{1}}-{{\mu }_{2}}\)
2. \({{\mu }_{D}}\)
3. \(\mu \)
4. \({{\mu }_{1}}-{{\mu }_{2}}\)
5. \({{\mu }_{D}}\)

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Solution: The downloadable solution consists of 7 pages, 482 words.
Deliverable: Word Document