Improve Your Understanding of College Level Statistics Problems

May and Adam are statistics students and reporters for their high school paper. In preparation for an upcoming article, they asked a random sample of students, "What do you look for in a significant other?" The table shows the responses broken down by the respondents' current relationship status. When analyzing their data, which test would be appropriate for these student reporters to use?
$\begin{array}{cc}& \text{Responses}\\ \text{Single?}& \begin{array}{|cccccc|}\hline & \text{ Looks }& \text{ Humor }& \text{ Smarts }& \text{ Money }& \text{ Kindness }\\ \text{ Yes }& 18& 17& 12& 8& 15\\ \text{ No }& 15& 19& 18& 4& 19\\ \hline\end{array}\end{array}$
Chi-square test of/for:
a. association/independence
b. goodness-of-fit
c. homogeneity

Jane has scores of 84, 92, and 89 on her first 3 exams. What does she need to score on the next exam so that her overall average will be 90?

For the following situations, identify the test you would run to analyze the data:
A marketing firm producing costumes is interested in studying consumer behavior in the context of purchase decision of costumes in a specific market. This company is a major player in the costume market that is characterized by intense competition. The company would like to know in particular whether the income level of the consumers (measured as lower, middle, upper middle, and upper class) influences their choice of costume type. They are specifically focused on four types of costumes (funny costumes, scary costumes, clever costumes, and boring costumes).
a.Chi-Square Goodness of Fit
b.Frequencies
c.Descriptive Statistics
d.Chi-Square of Independence

Does chocolate help heart-attack victims live longer? Researchers in Sweden randomly selected 1169 people who had suffered heart attacks and asked them about their consumption of chocolate in the previous year. Then the researchers followed these people and recorded whether or not they had died within 8 years.
... for goodness of fit, homogeneity or independence?
Who knows?

Do types of crimes reported on television newscasts differ by the age of the criminals? A researcher uses her dvr to record the local television news for a 2-week period of time. She records the types of crimes and notes the age of the offender. Conduct a chi-square test (${\displaystyle \alpha =0.05}$) of the null hypothesis that televised offense type does not vary by age group of offender.
$\begin{array}{|cccc|}\hline \text{ Offense type }& \text{ Young }& \text{ middle age }& \text{ older }\\ \text{ violent }& 27& 10& 3\\ \text{ property }& 14& 12& 11\\ \text{ sex }& 7& 9& 5\\ \hline\end{array}$
Report the obtained chi-square, df, and critical chi-square. Report your findings based on the null hypothesis.

Discuss the following;
What type of data can be examined using the Chi-Squared Test?
What is the only constraint in using Chi-Squared Tests?
What problems can be caused by the way the data have been grouped?

The students in an Introductory Statistics class are asked to complete a survey about study habits.
(a) does the study appear to use random sampling?
yes or no
(b) does the study appear to use random assignment?
yes or no

How do you find the median of 33 35 31 30 17 4 18 1 11 19 39 6 14 20?

Use Table VII to find the required ${\displaystyle {\chi }^2}$-values. Illustrate your work graphically.
When you use chi-square procedures to make inferences about a population standard deviation, why should the variable under consideration be normally distributed or nearly so?

Use ${\displaystyle 2\sum _{i=1}^n\frac{Y_i}{\beta }}$ which is a pivotal quantity to derive a ${\displaystyle 95\mathrm{\%}}$ confidence interval for ${\displaystyle \beta }$

Two-sided confidence intervals and tests
From a sample of 1751 army hospitals, estimate the mean expenses for a full time equivalent employee for all US army hospitals using a ${\displaystyle 90\mathrm{\%}}$ confidence interval given ${\displaystyle x=6563}$ and ${\displaystyle s=2484}$.

The table below shows the number of students absent from a college on particular days in a semester.
$\begin{array}{|ccccccc|}\hline \text{ Day }& \text{ Sun }& \text{ Mon }& \text{ Tue }& \text{ Wed }& \text{ Thu }& \text{ Total }\\ \text{ Frequency }& 100& 80& 87& 103& 110& 480\\ \hline\end{array}$
Use chi-square goodness of fit test to test the claim that the absence is distributed equally among the days at 0.05 level of significance.

How do you find the median of a set of values when there is an even number of values?

To test if TV preference and gender are independent, which hypothesis test would be used?
a. 2 Prop Z Test
b. Chi-square test
c. ! Sample T test
d. Linear Regression Test

The Pour It On marketing company is investigating whether the type of house and the color of a house in a certain town are related. Assume the conditions for inference are met.
Which of the following significance tests would be the most appropriate to use to determine whether there is an association between type of house and color of a house?
a. A two-sample t-test for mean.
b. A two sample z-test for mean.
c. Aregression t-test.
d. A chi-square goodness-of-fit test.
e. A chi-square test of independence.

The number of degrees of freedom associated with the chi-square distribution in a test of independence is
a. number of sample items minus 1.
b. number of populations minus number of estimated parameters minus 1.
c. number of populations minus 1.
d. number of rows minus 1 times number of columns minus 1.

Identify an appropriate statistical test and the tail of the test for the following problems. Assume all samples used for the study come from a normal population.
A. Determine if the performance of ABCS 206 students in an interim assessment(IA) is a good predictor of their performance at the end of semester examination.
B. A hair cream manufacturer wants to know if there is a relationship between the quantity of hair cream applied to the hair and the growth of hair.
C. A district health officer wishes to determine if the six villages in his district are independent of residents contracting malaria, schistosomiasis, cholera or tuberculosis.
D. A biologist is studying three different varieties of tomato to determine whether there is a difference in the proportion of seeds that germinate. Random samples of 200 seeds of each of three varieties are subjected to the same starting conditions.

How to keep the information of the confidence interval in the predictive step?
Suppose I have a sample ${\displaystyle X_1,X_2,\dots X_n}$. Each ${\displaystyle X_i}$ comes from a poisson distribution ${\displaystyle X_i\sim Po\left(\lambda \right)}$. The unknown paremeter ${\displaystyle \lambda }$ is inferred using the next confidence interval: ${\displaystyle {\lambda }^{\pm }=\bar{X}\pm t^{\cdot }\frac{\bar{X}}{\sqrt{n}}}$ where ${\displaystyle {\lambda }^{+}}$ is the upper bound and ${\displaystyle {\lambda }^{-}}$ is the lower bound, t∗ is the confidence expressed over t-student distribution.
I have two questions. The second question is actually the question I ask, the first is only to check my reasoning is right.
First: Taking into account the data come from Poisson distribution and taking into account the Central Limit Theorem, is it right the inference a did?It means: is it right to use the t-student approach? Or is it another better way to do?
Second: (That's the main question):
As I don't know the ${\displaystyle \lambda }$ parameter, I inferred it using confident intervals. Now I want to carry out some predictions of future events and calculate the associated probabilities, it means I wanto to know:
${\displaystyle prob\{X_{n+1}=K\}}$. To do this: What number in the interval ${\displaystyle [{\lambda }^{-},{\lambda }^{+}]}$ I should take? Obviously, maximize the likelihood is the best option, so I would take ${\displaystyle \bar{X}}$. So the question is:
Is there a way to take into account the uncertainty in the probabilities of the prediction? or: Is there a way to keep the knowledge of the confident interval in the probabilities of the new prediction? or: At the moment in that you use an only number(point estimation), how to keep the uncertainty information?
To be clearer take into account these two examples. The same process but with two different interval confidence:
1) ${\displaystyle [{\lambda }^{-},{\lambda }^{+}]=[6,8]}$
${\displaystyle \bar{X}=7\to X_{n+1}\sim Po(\lambda =\bar{X})\to Prob\{X_{n+1}=6\}=0.449}$
2) ${\displaystyle [{\lambda }^{-},{\lambda }^{+}]=[1,13]}$
${\displaystyle \bar{X}=7\to X_{n+1}\sim Po(\lambda =\bar{X})\to Prob\{X_{n+1}=6\}=0.449}$
The first contain less uncertainty than the second, so how could I maintain the information of confident interval?
I hope I was clear in the question.

There are 209 dogs who participate in a training to become service dogs.
There are four breeds of dogs.
Some dogs pass the certification test and some do not?
Chi square question:
Is there an association between breed of dog and passing the test? Or, are breed and passing the test independent of each other?
Blank tables are provided for your calculations.
1. What are the null and alternative hypotheses for the chi square test of independence?
2. Calculate the expected frequencies for each “cell”
3. Using the steps shown in the lecture and in the text, calculate the chi square to investigate whether “passed certification” and “breed of dog” are independent of each other or not.
4. Compare your calculated chi square to the critical value of chi square per the table, using the correct degrees of freedom.
$\begin{array}{|cccccc|}\hline & \text{ Chihuahua }& \text{ French Bulldog }& \text{ Golden Retriever }& \text{ Terriers }& \text{ Total }\\ \text{ Trained and certified as therapy dog }& 20& 12& 50& 37& 119\\ \text{ Trained but not certified as therapy dog }& 15& 15& 37& 23& 90\\ \text{ Total who took the training }& 35& 27& 87& 60& 209\\ \hline\end{array}$