# Baseball star David Ortiz-nicknamed "Big Papi"-is known for his ability to deliver hits in high-pressure situations. Here is a two-way table of his hits, walks, and outs in all of his regular-season and post-season plate appearances from 1997 through 2014. Choose a plate appearance at random. Are the events "Hit" and "Post-season" independent? Justify your answer.

Question
Two-way tables
Baseball star David Ortiz-nicknamed "Big Papi"-is known for his ability to deliver hits in high-pressure situations. Here is a two-way table of his hits, walks, and outs in all of his regular-season and post-season plate appearances from 1997 through 2014. Choose a plate appearance at random. Are the events "Hit" and "Post-season" independent? Justify your answer.

2021-03-13

DEFINITIONS Two events are independent, if the probability that one event occurs in no way affects the probability of the other event occurring. Definition conditional probability: $$\displaystyle{P}{\left({A}{\mid}{B}\right)}={\frac{{{P}{\left({A}\cap{B}\right)}}}{{{P}{\left({A}\right)}}}}={\frac{{{P}{\left({A}{\quad\text{and}\quad}{B}\right)}}}{{{P}{\left({A}\right)}}}}$$ SOLUTION We note that the table contains information about 8883 at-bats (given in the bottom right corner of the table). Moreover, 2110 of the 8883 at-bats are Hits, because 2110 is mentioned in the row ” Total” and in the column ” Hit” of the table. The probability is the number of favorable outcomes divided by the number of possible outcomes: $$P(Hit) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes} = \frac{2110}{8883} \approx 0.2375 = 23.75\%$$
Next, we note that 352 of the 8883 at-bats are post-season, because 352 is mentioned in the row ”Post” and in the column ”Total” of the given table. $$P(Post) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes}= \frac{352}{8883}$$
Next, we note that 87 of the 8883 at-bats are hits in post-season, because 87 is mentioned in the row ”Post” and in the column ”Hit” of the given table. $$P(Hit and Post) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes} = \frac{87}{8883}$$ Use the definition of conditional probability: $$\displaystyle{P}{\left({H}{i}{t}{\mid}{P}{o}{s}{t}\right)}={\frac{{{P}{\left({H}{i}{t}{\quad\text{and}\quad}{P}{o}{s}{t}\right)}}}{{{P}{\left({P}{o}{s}{t}\right)}}}}$$
$$\displaystyle{\frac{{\frac{{87}}{{8883}}}}{{\frac{{352}}{{8883}}}}}$$
$$\displaystyle{\frac{{{87}}}{{{352}}}}$$
$$\displaystyle\approx{0.2472}$$
$$\displaystyle={24.72}\%$$
If events A and B are independent, then $$P(A|B) = P(A)$$ and $$P(B|A) = P(B)$$
In this case, we note that $$P(Hit|Post) = 0.2472$$ is not the same as $$P(Hit) = 0.2375$$ and thus the two events are not independent.

### Relevant Questions

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. 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 known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think are the chances you will have much more than a middle-class income at age 30? The two-way table summarizes the responses.
$$\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Given that the chosen student didn't say "almost no chance," what's the probability that this person is female? Write your answer as a probability statement using correct symbols for the events.
1950 randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two-way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same as, or worse off than their parents
$$\begin{array}{|c|c|c|}\hline &\text{Less Than High School}&\text{High School}&\text{More Than High School}\\\hline \text{Better off} &140&440&430\\ \hline \text{Same as}&60&230&110\\ \hline \text{Worse off}&180&280&80\\ \hline\end{array}\\$$
Suppose one adult is selected at random from these 1950 adults. Find the following probablity.
$$P(\text{more than high school or worse off})=?$$
How are the smoking habits of students related to their parents' smoking? Here is a two-way table from a survey of student s in eight Arizona high schools:
$$\begin{array}{c|c}&\text{Student smokes}&\text{Student does not smoke}&\text{Total}\\\hline\text{Both parents smoke}&400&1380&400+1380=1780\\\hline\text{One parent smokes}&416&1823&416+1823=2239\\\hline\text{Neither parent smokes}&188&1168&188+1168=1356\\\hline\text{Total}&400+416+188=1004&1380+1823+1168=4371&1004+4371=5375\end{array}$$
(a) Write the null and alternative hypotheses for the question of interest.
(b) Find the expected cell counts. Write a sentence that explains in simple language what "expected counts" are.
(c) Find the chi-square statistic, its degrees of freedom, and the P-value.
Ivy conducted a taste test for 4 different brands of chocolate chip cookies. Here is a two-way table that describes which cookie each subject preferred and that person's gender.
$$\begin{array}{c|cccc} &A& B & C & D \\ \hline Yes &4& 6 & 13 & 13\\ No &22& 11 & 11 & 14\\ \end{array}$$
Suppose one subject from this experiment is selected at random. Find the probability that the selected subject preferred Brand C.
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.
$$\begin{array}{c|c} & 4th\ grade & 5th\ grade & 6th\ grade &Total \\ \hline Grades &49&50&69&168\\ Athletic &24&36&38&98\\ Popular\ &19&22&28&69\\ \hline Total & 92 & 108 & 135 &335 \end{array}$$
Suppose we select one of these students at random. What's the probability of each of the following? The student is not a sixth-grader and did not rate good grades as important.
The top string of a guitar has a fundamental frequency of 33O Hz when it is allowed to vibrate as a whole, along all its 64.0-cm length from the neck to the bridge. A fret is provided for limiting vibration to just the lower two thirds of the string, If the string is pressed down at this fret and plucked, what is the new fundamental frequency? The guitarist can play a "natural harmonic" by gently touching the string at the location of this fret and plucking the string at about one sixth of the way along its length from the bridge. What frequency will be heard then?
A child is playing on the floor of a recreational vehicle (RV) asit moves along the highway at a constant velocity. He has atoy cannon, which shoots a marble at a fixed angle and speed withrespect to the floor. The cannon can be aimed toward thefront or the rear of the RV. Is the range toward the frontthe same as, less than, or greater than the range toward the rear?Answer this question (a) from the child's point of view and (b)from the point of view of an observer standing still on the ground.Justify your answers.
The following table gives a two-way classification of all basketball players at a state university who began their college careers between 2004 and 2008, based on gender and whether or not they graduated.
$$\begin{array}{|c|c|c|}\hline &\text{Graduated}&\text{Did not Graduate}\\\hline \text{Male} &129&51\\ \hline \text{Female}&134&36 \\ \hline \end{array}\\$$
If one of these players is selected at random, find the following probability.
$$P(\text{graduated or male})=$$ Enter your answer in accordance to the question statement
$$\begin{array}{c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text{ Yes } & 19 & 15 & 34 \\ \text{ No } & 24 & 30 & 54 \\ \hline \text{ Total } & 43 & 45 & 88\\ \end{array}$$
$$(a)\frac{19}{88}$$
(b)$$\frac{39}{88}$$
(c)$$\frac{58}{88}$$
(d)$$\frac{77}{88}$$