# The accompanying two-way table was constructed using data in the article “Television Viewing and Physical Fitness in Adults” (Research Quarterly for Exercise and Sport, 1990: 315–320). The author hoped to determine whether time spent watching television is associated with cardiovascular fitness. Subjects were asked about their television-viewing habits and were classified as physically fit if they scored in the excellent or very good category on a step test. We include MINITAB output from a chi-squared analysis. The four TV groups corresponded to different amounts of time per day spent watching TV (0, 1–2, 3–4, or 5 or more hours). The 168 individuals represented in the first column were those judged physically fit. Expected counts appear below observed counts, and MINITAB displays the con Question
Two-way tables The accompanying two-way table was constructed using data in the article “Television Viewing and Physical Fitness in Adults” (Research Quarterly for Exercise and Sport, 1990: 315–320). The author hoped to determine whether time spent watching television is associated with cardiovascular fitness. Subjects were asked about their television-viewing habits and were classified as physically fit if they scored in the excellent or very good category on a step test. We include MINITAB output from a chi-squared analysis. The four TV groups corresponded to different amounts of time per day spent watching TV (0, 1–2, 3–4, or 5 or more hours). The 168 individuals represented in the first column were those judged physically fit. Expected counts appear below observed counts, and MINITAB displays the contribution to $$\displaystyle{x}^{{{2}}}$$ from each cell.
State and test the appropriate hypotheses using $$\displaystyle\alpha={0.05}$$
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}&{a}\mp,\ {1}&{a}\mp,\ {2}&{a}\mp,\ {T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{1}&{a}\mp,\ {35}&{a}\mp,\ {147}&{a}\mp,\ {182}\backslash{h}{l}\in{e}&{a}\mp,\ {25.48}&{a}\mp,\ {156.52}&{a}\mp,\backslash{h}{l}\in{e}{2}&{a}\mp,\ {101}&{a}\mp,\ {629}&{a}\mp,\ {730}\backslash{h}{l}\in{e}&{a}\mp,\ {102.20}&{a}\mp,\ {627.80}&{a}\mp,\backslash{h}{l}\in{e}{3}&{a}\mp,\ {28}&{a}\mp,\ {222}&{a}\mp,\ {250}\backslash{h}{l}\in{e}&{a}\mp,\ {35.00}&{a}\mp,\ {215.00}&{a}\mp,\backslash{h}{l}\in{e}{4}&{a}\mp,\ {4}&{a}\mp,\ {34}&{a}\mp,\ {38}\backslash{h}{l}\in{e}&{a}\mp,\ {5.32}&{a}\mp,\ {32.68}&{a}\mp,\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{a}\mp,\ {168}&{a}\mp,\ {1032}&{a}\mp,\ {1200}\backslash{h}{l}\in{e}$$
$$\displaystyle{C}{h}{i}{s}{q}={a}\mp,\ {3.557}\ +\ {0.579}\ +\ {a}\mp,\ {0.014}\ +\ {0.002}\ +\ {a}\mp,\ {1.400}\ +\ {0.228}\ +\ {a}\mp,\ {0.328}\ +\ {0.053}={6.161}$$
$$\displaystyle{d}{f}={3}$$ 2020-11-24
Step 1
Testing for Independence - Lack of Association
When testing null hypothesis
$$\displaystyle{H}_{{{0}}}\ :\ {p}_{{{i}{j}}}={p}_{{{i}.}}\ \cdot\ {p}_{{.{j}}},\ {i}={1},\ {2},\ \cdots\ ,{I},\ {j}={1},\ {2},\ \cdots\ ,{J}$$
versus alternative hypothesis
$$\displaystyle{H}_{{\alpha}}\ :\ {H}_{{{0}}}$$ is not true.
Let $$\displaystyle\hat{{{e}}}_{{{i}{j}}}\ \geq\ {5}\ {f}{\quad\text{or}\quad}\ {e}{a}{c}{h}\ {i},\ {j},\ {w}{h}{e}{r}{e}\ \hat{{{e}}}_{{{i}{j}}}$$
$$\displaystyle\hat{{{e}}}_{{{i}{j}}}={n}\ \cdot\ \hat{{{p}}}_{{{i}.}}\ \cdot\ \hat{{{p}}}_{{.{j}}}={n}\ \cdot\ {\frac{{{n}_{{{i}.}}}}{{{n}}}}\ \cdot\ {\frac{{{n}_{{.{j}}}}}{{{n}}}}={\frac{{{n}_{{{i}.}}\ \cdot\ {n}_{{.{j}}}}}{{{n}}}}$$
under regularity conditions, test statistic value is
$$\displaystyle{X}^{{{2}}}={\sum_{{{j}={1}}}^{{{I}}}}\ {\sum_{{{i}={1}}}^{{{J}}}}\ {\frac{{{\left({n}_{{{i}{j}}}\ -\ \hat{{{e}}}_{{{i}{j}}}\right)}^{{{2}}}}}{{\hat{{{e}}}_{{{i}{j}}}}}}=\ \sum_{{{a}{l}{l}\ {c}{e}{l}{l}{s}}}\ {\frac{{{o}{b}{s}{e}{r}{v}{e}{d}-{e}{s}{t}{i}{m}{a}{t}{e}{d}\ {\exp{{e}}}{c}{t}{e}{d}}}{^}}{\left\lbrace{2}\right\rbrace}\rbrace{\left\lbrace{e}{s}{t}{i}{m}{a}{t}{e}{d}\ {\exp{{e}}}{c}{t}{e}{d}\right\rbrace}$$
has approximated a chi-square distribution with (I - 1)(J - 1) degrees of freedom when $$\displaystyle{H}_{{{0}}}$$ is true.
The P-value is corresponding area to the right of $$\displaystyle{X}^{{{2}}}\ {u}{n}{d}{e}{r}\ {t}{h}{e}\ {{X}_{{{\left({I}\ -\ {1}\right)}{\left({J}\ -\ {1}\right)}}}^{{{2}}}}$$ curve.
The null hypothesis is
$$\displaystyle{H}_{{{0}}}\ :$$ Television viewing and physical fitness are independent
versus alternative
$$\displaystyle{H}_{{\alpha}}\ :$$ Television viewing and pshysical fitness are not independent.
Critical value, from the table in the appendix, is given by
$$\displaystyle{{X}_{{{0.05},\ {\left({4}\ -\ {1}\right)}{\left({2}\ -\ {1}\right)}}}^{{{2}}}}={7.815},$$
and the calculated $$\displaystyle{X}^{{{2}}}$$ is given in the output as
$$\displaystyle{X}^{{{2}}}={3.557}\ +\ {0.579}\ +\ {0.014}\ +\ {0.002}\ +\ {1.400}\ +\ {0.228}\ +\ {0.328}\ +\ {0.053}={6.161},$$
thus, since
$$\displaystyle{{X}_{{{0.05},\ {3}}}^{{{2}}}}={7.815}\ {>}\ {6.161}={X}^{{{2}}},$$
do not reject null hypothesis at given significance level. The data indicates that there is no association between variables.

### Relevant Questions n an experiment designed to study the effects of illumination level on task performance (“Performance of Complex Tasks Under Different Levels of Illumination,” J. Illuminating Eng., 1976: 235–242), subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with a black background and a higher level with a white background. Each data value is the time (sec) required to complete the task. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\mathcal}\right\rbrace}{h}{l}\in{e}&{a}\mp&{a}\mp&{a}\mp\ \text{Subject}\backslash{h}{l}\in{e}&{a}\mp\ {1}&{a}\mp\ {2}&{a}\mp\ {3}&{a}\mp\ {4}&{a}\mp\ {5}&{a}\mp\ {6}&{a}\mp\ {7}&{a}\mp\ {8}&{a}\mp\ {9}&{a}\mp\backslash{h}{l}\in{e}\text{Black}&{a}\mp\ {25.85}&{a}\mp\ {28.84}&{a}\mp\ {32.05}&{a}\mp\ {25.74}&{a}\mp\ {20.89}&{a}\mp\ {41.05}&{a}\mp\ {25.01}&{a}\mp\ {24.96}&{a}\mp\ {27.47}&{a}\mp\backslash{h}{l}\in{e}\text{White}&{a}\mp\ {18.23}&{a}\mp\ {20.84}&{a}\mp\ {22.96}&{a}\mp\ {19.68}&{a}\mp\ {19.509}&{a}\mp\ {24.98}&{a}\mp\ {16.61}&{a}\mp\ {16.07}&{a}\mp\ {24.59}&{a}\mp\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Does the data indicate that the higher level of illumination yields a decrease of more than 5 sec in true average task completion time? Test the appropriate hypotheses using the P-value approach. In an experiment designed to study the effects of illumination level on task performance (“Performance of Complex Tasks Under Different Levels of Illumination,” J. Illuminating Eng., 1976: 235–242), subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with a black background and a higher level with a white background. Each data value is the time (sec) required to complete the task.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{S}{u}{b}{j}{e}{c}{t}&{\left({1}\right)}&{\left({2}\right)}&{\left({3}\right)}&{\left({4}\right)}&{\left({5}\right)}&{\left({6}\right)}&{\left({7}\right)}&{\left({8}\right)}&{\left({9}\right)}\backslash{h}{l}\in{e}{B}{l}{a}{c}{k}&{25.85}&{28.84}&{32.05}&{25.74}&{20.89}&{41.05}&{25.01}&{24.96}&{27.47}\backslash{h}{l}\in{e}{W}{h}{i}{t}{e}&{18.28}&{20.84}&{22.96}&{19.68}&{19.509}&{24.98}&{16.61}&{16.07}&{24.59}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Does the data indicate that the higher level of illumination yields a decrease of more than 5 sec in true average task completion time? Test the appropriate hypotheses using the P-value approach. 1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance. Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. This two-way table summarizes the survey data. PSK\begin{array} {lc} & Grade \ Most important & \begin{array}{l|c|c|c|c} & \begin{array}{c} 4 \mathrm{th} \ grade \end{array} & \begin{array}{c} 5 \mathrm{th} \ \text { grade } \end{array} & \begin{array}{c} 6 \mathrm{th} \ grade \end{array} & Total \\ \hline Grades & 49 & 50 & 69 & 168 \\ \hline Athletic & 24 & 36 & 38 & 98 \\ \hline Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array} \ \end{array}ZSK Suppose we select one of these students at random. What's the probability that: The student is a sixth grader or a student who rated good grades as important? A random sample of 2,500 people was selected, and the people were asked to give their favorite season. Their responses, along with their age group, are summarized in the two-way table below.
$$\begin{array}{c|cccc|c} & \text {Winter} &\text{Spring}& \text {Summer } & \text {Fall}& \text {Total}\\ \hline \text {Children} & 30 & 0 & 170&0&200 \\ \text{Teens} & 150 & 75 & 250&25&500 \\ \text {Adults } & 250 & 250 & 250&250&1000 \\ \text {Seniors} & 300 & 150 & 50&300&800 \\ \hline \text {Total} & 730 & 475 & 720 &575&2500 \end{array}$$
Among those whose favorite season is spring, what proportion are adults?
$$a) \frac{250}{1000}$$
$$b) \frac{250}{2500}$$
$$c) \frac{475}{2500}$$
$$d) \frac{250}{475}$$
$$e) \frac{225}{475}$$ Iron is very important for babies' growth. A common belief is that breastfeeding will help the baby to get more iron than formula feeding. To justify the belief, a study followed 2 groups of babies from born to 6 months. With one group babies are breast fed, and the other group are formula fed without iron supplements. Data below shows iron levels of those two groups of babies. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{G}{r}{o}{u}{p}&{S}{a}\mp\le\ {s}{i}{z}{e}&{m}{e}{a}{n}&{S}{\tan{{d}}}{a}{r}{d}\ {d}{e}{v}{i}{a}{t}{i}{o}{n}\backslash{h}{l}\in{e}{B}{r}{e}\ast-{f}{e}{d}&{23}&{13.3}&{1.7}\backslash{h}{l}\in{e}{F}{\quad\text{or}\quad}\mu{l}{a}-{f}{e}{d}&{23}&{12.4}&{1.8}\backslash{h}{l}\in{e}{D}{I}{F}{F}={B}{r}{e}\ast-{F}{\quad\text{or}\quad}\mu{l}{a}&{23}&{0.9}&{1.4}\backslash{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ (1) There are two groups we need to compare for the study: Breast-Fed and Formula- Fed. Are those two groups dependent or independent? Based on your answer, what inference procedure should we apply for this research? (2) Please perform the inference you decided in (1), and make sure to follow the 5-step procedure for any hypothesis test. (3) Based on your conclusion in (2), what kind of error could you make? Explain the type of error using the context words for this research Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data. $$\displaystyle{G}{r}{a}{d}{e}\ {M}{o}{s}{t}{i}\mp{\quad\text{or}\quad}{\tan{{t}}}{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{\left|{c}\right|}{c}{\left|{c}\right|}{c}\right\rbrace}&{4}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}&{5}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}&{6}{m}{a}{t}{h}{r}{m}{\left\lbrace{t}{h}\right\rbrace}&\ \text{ Total }\ \backslash{h}{l}\in{e}{G}{r}{a}{d}{e}{s}&{49}&{50}&{69}&{168}\backslash{h}{l}\in{e}{A}{t}{h}\le{t}{i}{c}&{24}&{36}&{38}&{98}\backslash{h}{l}\in{e}{P}{o}{p}\underline{{a}}{r}&{19}&{22}&{28}&{69}\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{92}&{108}&{135}&{335}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Suppose we select one of these students at random. Find P(athletic | 5th grade). A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared. In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample.
This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples.
The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination. PSK\begin{array}{|c|c|} \hline Geographical\ regions & Before\ vaccination & After\ vaccination\\ \hline 1 & 85 & 11\\ \hline 2 & 77 & 5\\ \hline 3 & 110 & 14\\ \hline 4 & 65 & 12\\ \hline 5 & 81 & 10\\\hline 6 & 70 & 7\\ \hline 7 & 74 & 8\\ \hline 8 & 84 & 11\\ \hline 9 & 90 & 9\\ \hline 10 & 95 & 8\\ \hline \end{array}ZSK
Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Construct a one-sided $$\displaystyle{95}\%$$ confidence interval for the true difference in population means. Test the null hypothesis that the population means are identical at the 0.05 level of significance. $$\begin{array}{c|c}&\text{Student smokes}&\text{Student does not smoke}&\text{Total}\\\hline\text{Both parents smoke}&400&1380&400+1380=1780\\\hline\text{One parent smokes}&416&1823&416+1823=2239\\\hline\text{Neither parent smokes}&188&1168&188+1168=1356\\\hline\text{Total}&400+416+188=1004&1380+1823+1168=4371&1004+4371=5375\end{array}$$ A random sample of U.S. adults was recently asked, "Would you support or oppose major new spending by the federal government that would help undergraduates pay tuition at public colleges without needing loans?" The two-way table shows the responses, grouped by age. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\mathcal{{c}}}\right\rbrace}&{A}\ge\ {R}{e}{s}{p}{o}{n}{s}{e}&{\left\lbrace{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{\left|{r}\right|}{r}{\left|{r}\right|}{r}{\mid}{r}\right\rbrace}&{18}-{34}&{35}-{49}&{50}-{64}&{65}+&{T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{S}{u}{p}{p}{\quad\text{or}\quad}{t}&{91}&{161}&{272}&{332}&{856}\backslash{h}{l}\in{e}{O}{p}{p}{o}{s}{e}&{25}&{74}&{211}&{255}&{565}\backslash{h}{l}\in{e}{D}{o}{n}'{t}{k}{n}{o}{w}&{4}&{13}&{20}&{51}&{88}\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{120}&{248}&{503}&{638}&{1509}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right\rbrace}\ {e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Do these data provide convincing evidence of an association between age and opinion about loan-free tuition in the population of U.S. adults?