6.4.1: Exercises
- Page ID
- 48855
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Pearl wonders if the majority of US adults are dissatisfied with the quality of the environment. According to a poll conducted by Gallup10 of 200 randomly surveyed US adults, 122 respondents were dissatisfied with the quality of the environment. Test Pearl’s claim at a 5% level of significance.
- \(p\) represents the proportion of US adults who are:
- \(H_0\):
- \(H_a\):
- What test should you use to find the P-value? Justify your answer.
- Explain why the sampling distribution of sample proportions is approximately normal.
- What is the sample proportion, \(\hat{p}\)? Write your answer as a fraction and decimal.
- Compute the Z-score for the sample proportion.
- Use desmos to find the P-value from the standard normal distribution. Sketch a graph and shade the area that represents the P-value.
- Make a decision about the null and alternative hypotheses. Justify your answer.
- State the conclusion in context.
- \(p\) represents the proportion of US adults who are:
- In a random sample of 300 Alzheimer’s patients taking a new drug, 21 experienced nausea as a side effect. The drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer’s disease will experience nausea. Test the claim at a 1% level of significance.
- Step 1
- Step 2
- Step 3
- Step 4
- Step 1
- The proportion of smokers among persons who graduated from a four-year university has been widely reported as 22%. A sociologist student wonders if this is still true. They randomly sample 785 four-year university graduates and finds that 157 are smokers. They test the claim at a 5% level of significance. Spot the errors in the students solution below (there is at least one error in each step):
Step 1: \(p\) represents four-year university graduates who are smokers.
\(H_0: \hat{p}=0.22\)
\(H_a: \hat{p} \neq 0.22\)
Step 2: There are 157 successes in the sample and 785-157=628 failures in the sample. These are greater than 10 so it’s normal. \(\hat{p}=\frac{157}{785}=0.2\).
Step 3: \(Z=\frac{0.2-0.22}{\sqrt{\frac{0.2(1-0.2)}{785}}} \approx-1.40\)
Images are created with the graphing calculator, used with permission from Desmos Studio PBC.
The P-value is 0.0808.
Step 4: The P-value 0.0808 is greater than the level of significance 0.05. We accept the null hypothesis and reject the alternative hypothesis.
The sample data show that the four-year university graduates who are smokers is equal to 22%.
- The proportion of smokers among persons who graduated from a four-year university has been widely reported as 22%. A sociologist student wonders if this is still true. They randomly sample 785 four-year university graduates and finds that 157 are smokers. Test their claim at a 5% level of significance. Clearly show each step of a hypothesis test.
Reference
10Jeffrey M. Jones, “Americans Offer Gloomy State of the Nation Report,” Gallup.com, February 2, 2022, accessed September 27, 2022, https://news.gallup.com/poll/389309/americans-offer-gloomy-state-nation-report.aspx