# The tables show the battery lives (in hours) of two brands of laptops. a) Make a double box-and-whisker plot that represent's the data. b) Identifity the shape of each distribution. c) Which brand's battery lives are more spread out? Explain. d) Compare the distributions using their shapes and appropriate measures of center and variation.

Question
Modeling data distributions
The tables show the battery lives (in hours) of two brands of laptops. a) Make a double box-and-whisker plot that represent's the data. b) Identifity the shape of each distribution. c) Which brand's battery lives are more spread out? Explain. d) Compare the distributions using their shapes and appropriate measures of center and variation.

2020-12-31
a) Double box-and-whisker plot: For brand A, Arranging the data in ascending order, $$\displaystyle{8.5},\ {11.5},\ {13.5},\ {13.5},\ {14.5},\ {15.5},\ {16.25},\ {16.75},\ {18.5},\ {20.75}$$
$$\displaystyle\text{Lowest value}={8.5}$$
$$\displaystyle\text{Highest value}={20.75}$$
$$\displaystyle\text{Lower quartile}={13.5}$$
$$\displaystyle\text{Upper quartile}={16.75}$$
$$\displaystyle\text{Median}={\frac{{{14.5}\ +\ {15.5}}}{{{2}}}}$$
$$\displaystyle={15}$$ For brand B, Arranging the data in ascending order, $$\displaystyle{7},\ {8.5},\ {8.5},\ {9},\ {9},\ {9.5},\ {9.75},\ {10.25},\ {12.5}$$
$$\displaystyle\text{Lowest value}={7.8}$$
$$\displaystyle\text{Highest value}={12.5}$$
$$\displaystyle\text{Lower quartile}={8.5}$$
$$\displaystyle\text{Upper quartile}={10.25}$$
$$\displaystyle\text{Median}={\frac{{{9}\ +\ {9.5}}}{{{2}}}}$$
$$\displaystyle={9.25}$$ b) For Brand A. The median is placed towards the left-side of the box, this means that most of the data lies on left side and distribution is skewed right. For Brand B. The median is placed towards the left-side of the box, this means that most of the data lies on left-side and distribution is skewed right. c) Brand A's battery lives are more spread out because its range $$\displaystyle{\left({20.75}\ -\ {8.5}={12.25}\right)}$$ is greater as compared to brand B's battery lives range $$\displaystyle{\left({12.5}\ -\ {7}={5.5}\right)}.$$ d) As both distributions are skewed, median should be used as a measure of center and five-number summary for measuring variation. For brand A, median is 15 while for brand B median is 9.25.

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$$\displaystyle{X}_{{1}}:$$ Rate of hay fever per 1000 population for people under 25
$$\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}$$
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$$\displaystyle{X}_{{2}}:$$ Rate of hay fever per 1000 population for people over 50
$$\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}$$
(i) Use a calculator to calculate $$\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.$$ (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use $$\displaystyle\alpha={0.05}.$$
(a) What is the level of significance?
State the null and alternate hypotheses.
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}$$
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value $$\displaystyle>{0.250}$$
$$\displaystyle{0.125}<{P}-\text{value}<{0},{250}$$
$$\displaystyle{0},{050}<{P}-\text{value}<{0},{125}$$
$$\displaystyle{0},{025}<{P}-\text{value}<{0},{050}$$
$$\displaystyle{0},{005}<{P}-\text{value}<{0},{025}$$
P-value $$\displaystyle<{0.005}$$
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
P-value f P-value
$$\begin{array}{c|c}&Yes&No\\\hline\text{Children}&0.15&0.25\\\hline\text{Adults}&0.1&0.6\end{array}$$
Let's say the widget maker has developed the following table that shows the highest dollar price p. widget where you can sell N widgets. Number N Price p $$200 53.00$$
$$250 52.50$$
$$300 52.00$$
$$35051.50$$ (a) Find a formula for pin terms of N modeling the data in the table. (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in month as a function of the number N of widgets produced in a month. $$R=$$ Is Ra linear function of N? (c) On the basis of the tables in this exercise and using cost, $$C= 35N + 900$$, use a formula to express the monthly profit P, in dollars, of this manufacturer asa function of the number of widgets produced in a month $$p=$$ (d) Is Pa linear function of N2 e. Explain how you would find breakeven. What does breakeven represent?