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A newspaper columnist writes that "a majority of Republicans believe that the 2020 election was stolen" What null hypothesis and alternative hypothesis should you use in testing that statement? \( \begin{array}{ll}\mathrm{H}_{0}: p>0.50 & \mathrm{H}_{\mathrm{a}}: p=0.50 \\ \mathrm{H}_{0}: p \neq 0.50 & \mathrm{H}_{\mathrm{a}}: p=0.50 \\ \mathrm{H}_{0}: p=0.50 & \mathrm{H}_{\mathrm{a}}: p>0.50 \\ \mathrm{H}_{0}: p<0.50 & \mathrm{H}_{\mathrm{a}}: p=0.50 \\ \mathrm{H}_{0}: p=0.50 & \mathrm{H}_{\mathrm{a}}: p \geq 0.50 \\ \mathrm{H}_{0}: p=0.50 & \mathrm{H}_{\mathrm{a}}: p \neq 0.50 \\ \mathrm{H}_{0}: p=0.50 & \mathrm{H}_{\mathrm{a}}: p \leq 0.50 \\ \mathrm{H}_{0}: p \neq 0.50 & \mathrm{H}_{\mathrm{a}}: p>0.50\end{array} \)

The mean weight of pills coming off a drug company's manufacturing line is supposed to be 3.7 grams. If a quality control engineer wants to test that the manufacturing line is working property, what null and altemative hypotheses should the engineer use? \( \begin{array}{ll}\mathrm{H}_{0}: \mu=3.7 & \mathrm{H}_{\mathrm{a}}: \mu \geq 3.7 \\ \mathrm{H}_{0}: \mu \neq 3.7 & \mathrm{H}_{\mathrm{a}}: \mu<3.7 \\ \mathrm{H}_{0}: \mu \neq 3.7 & \mathrm{H}_{\mathrm{a}}: \mu=3.7 \\ \mathrm{H}_{0}: \mu \geq 3.7 & \mathrm{H}_{\mathrm{a}}: \mu=3.7 \\ \mathrm{H}_{0}: \mu=3.7 & \mathrm{H}_{\mathrm{a}}: \mu>3.7 \\ \mathrm{H}_{0}: \mu<3.7 & \mathrm{H}_{\mathrm{a}}: \mu-3.7 \\ \mathrm{H}_{0}: \mu=3.7 & \mathrm{H}_{\mathrm{a}}: \mu \neq 3.7 \\ \mathrm{H}_{\mathrm{a}}: \mu=3.7 & \mathrm{H}_{\mathrm{a}}: \mu<3.7\end{array} \)

A college profersor reads a newspaper article that claims that \( 57 \% \) of college students are women. The profersor wonders if that's true. Find the null and alternative hypotheses that would be used to test the statement in the newspaper article. \( \begin{array}{ll}\mathrm{H}_{0}: p=0.57 & \mathrm{H}_{\mathrm{a}}: p \geq 0.57 \\ \mathrm{H}_{0}: p=0.57 & \mathrm{H}_{\mathrm{a}}: p \leq 0.57 \\ \mathrm{H}_{0}: p \neq 0.57 & \mathrm{H}_{\mathrm{a}}: p>0.57 \\ \mathrm{H}_{0}: p>0.57 & \mathrm{H}_{\mathrm{a}}: p=0.57 \\ \mathrm{H}_{0}: p=0.57 & \mathrm{H}_{\mathrm{a}}: p \neq 0.57 \\ \mathrm{H}_{0}: p \neq 0.57 & \mathrm{H}_{\mathrm{a}}: p=0.57 \\ \mathrm{H}_{0}: p<0.57 & \mathrm{H}_{\mathrm{a}}: p=0.57 \\ \mathrm{H}_{0}: p=0.57 & \mathrm{H}_{\mathrm{a}}: p>0.57\end{array} \)

A college professor reads a newspaper article that claims that \( 57 \% \) of college students are women. The professor thinks that the percentage is actually higher than that. Find the null and alternative hypotheses that would be used to test the professor's claim. \( \begin{array}{ll}\mathrm{H}_{0}: \mu=0.57 & \mathrm{H}_{\mathrm{a}}: \mu \neq 0.57 \\ \mathrm{H}_{0}: p \neq 0.57 & \mathrm{H}_{\mathrm{a}}: p=0.57 \\ \mathrm{H}_{0}: p=0.57 & \mathrm{H}_{\mathrm{a}}: p \neq 0.57 \\ \mathrm{H}_{0}: \mu \neq 0.57 & \mathrm{H}_{\mathrm{a}}: \mu=0.57 \\ \mathrm{H}_{0}: p=0.57 & \mathrm{H}_{\mathrm{a}}: p>0.57 \\ \mathrm{H}_{0}: p>0.57 & \mathrm{H}_{\mathrm{a}}: p=0.57 \\ \mathrm{H}_{0}: \mu=0.57 & \mathrm{H}_{\mathrm{a}}: \mu>0.57 \\ \mathrm{H}_{0}: \mu<0.57 & \mathrm{H}_{\mathrm{a}}: \mu=0.57\end{array} \)

A cereal manufacturer claims that the mean weight of cereal in its \( 12-\mathrm{o} \) boxes is at least 12.3 oz. Write the null and alternative hypotheses that would be used to test that claim. \( \mathrm{H}_{0}: \mu=12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu \neq 12.3 \) \( \mathrm{H}_{0}: \mu \neq 12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu=12.3 \) \( \mathrm{H}_{0}: \mu \neq 12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu<12.3 \) \( \mathrm{H}_{0}: \mu=12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu \geq 12.3 \) \( \mathrm{H}_{0}: \mu \geq 12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu=12.3 \) \( \mathrm{H}_{0}: \mu=12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu>12.3 \) \( \mathrm{H}_{0}: \mu=12.3 \quad \mathrm{H}_{\mathrm{a}}: \mu<12.3 \) \( \mathrm{H}_{0}: \mu<12.3 \) \( \mathrm{H}_{\mathrm{a}}: \mu=12.3 \)

\( Q_{A}:(a)- \) Prove that \( \operatorname{cov}\left(y, b_{1}\right)=0 \)

c. La cantidad de horas que un grupo de personas ven TV al día son \( 5,2,1,0,5,4 \), 3,2 y 2 . d. La cantidad de panes que consume un grupo de familias al día en el desayuno son \( 2,4,4,6,5 \) y 6 .

student would like to test that contribution of gender in wage is \( \mathbf{1 \$ 8} \) per hour and that contributi f one year of education is two times greater than one year of experience. What is the appropriate xpression for the joint hypothesis? a. \( \quad\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ 0 & 0 & -2 & 1\end{array}\right)\left(\begin{array}{l}\beta_{1} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4}\end{array}\right)=\binom{1}{0} \quad \) b. \( \quad\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right)\left(\begin{array}{l}\beta_{1} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4}\end{array}\right)=\left(\begin{array}{l}0 \\ 1 \\ 0 \\ 0\end{array}\right) \)

e. your answer. A student would like to test that contribution of gender in wage is 1 \$ per hour and that contributio of one year of education is two times greater than one year of experience. What is the appropriate expression for the joint hypothesis? a. \( \quad\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & -2 & 1\end{array}\right)\left(\begin{array}{l}\beta_{1} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4}\end{array}\right)=\binom{1}{0} \quad \) b. \( \quad\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right)\left(\begin{array}{l}\beta_{1} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4}\end{array}\right)=\left(\begin{array}{l}0 \\ 1 \\ 0 \\ 0\end{array}\right) \)

6 A sample of people were given two versions of a drink, the original recipe and a new recipe. They were asked, 'Do you prefer the new recipe?" \( 85 \% \) said, yes'. a Why might this result be biased? b How could you arrange the tasting and questioning to avoid bias?