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statistics
elementary statistics in social research

Questions and Answers of Elementary Statistics In Social Research

Derive the negative binomial (NB) distribution.(a) Suppose \(y\) follows a Poisson \((\lambda)\), where the parameter \(\lambda\) itself is a random variable following a gamma distribution
Prove the equation below (2.11). - P11 P1+P+1 ~a N (P11 P1+P+1, [P+P+1 (1 P1+) (1 P+1)]). (2.11) n
Consider the statistic in (2.14).(a) Show that this statistic is asymptotically normal with the asymptotic variance given
For the DOS, test whether education is associated with depression. To simplify the problem, we dichotomize both variables; use no and major/minor for depression diagnosis and at most and more than 12
Derive the relationships among the eight versions of odds ratios of Section 2.2.2.When twovariables(orrowandcolumn)areactuallyassociated,wemaywanttoknowthenature
Let \(p_{1}=\operatorname{Pr}(y=1 \mid x=1)=0.8\) and \(p_{2}=\operatorname{Pr}(y=1 \mid x=0)=0.4\).(a) Compute the relative risk of response \(y=1\) of population \(x=1\) to population \(x=0\), and
Show that the hypergeometric distribution \(H G\left(k ; n, n_{1+}, n_{+1}\right)\) has mean \(\frac{n_{1+} n_{+1}}{n}\) and variance \(\frac{n_{1+} n_{+1} n_{+2} n_{2+}}{n^{2}(n-1)}\).
In the PPD, each subject was diagnosed for depression using SCID along with several screening tests including EPDS. By repeatedly dichotomizing the EPDS outcome, answer the following questions:(a)
The data set "intake" contains baseline information of the Catheter Study. Use the two binary outcomes on whether urinary tract infections (UTIs) and catheter blockages occurred during the last two
Group the count responses on UTIs and catheter blockages in the data set "intake" into three levels: no occurrence, only once, and more than once. Use these three-level outcomes to assess(a) whether
For the DOS, use the three-level depression diagnosis and dichotomized education (more than 12 years education or not) to check the association between education and depression.(a) Test whether
The data set "DosPrepost" contains depression diagnosis of patients at baseline (pretreatment) and one year after treatment (posttreatment) in the DOS. We are interested in whether there is any
Let \(p\) denote the prevalence of a disease of interest. Express \(P P V\) and \(N P V\) as a function of \(p, S e\), and \(S p\).
Prove that the weighted kappa for \(2 \times 2\) tables will reduce to the simple kappa, no matter which weights are assigned to the two levels.
Verify the variance formula for the MWW statistics (2.29). nin2 (N+1) 12 r 12N (N-1) (n+j - 1) n+j (n+j+1) (2.29) j=1
Use the three-level depression diagnosis and dichotomized education (more than 12 years education or not) in the DOS data to test the association between education and depression.(a) Use the Pearson
For the \(2 \times r\) table with scores as in Section 2.3.1,(a) verify that the MLE of \(\beta\) in the linear regression model in (2.26) is \(\widehat{\beta}=\frac{\overline{x y}-\bar{x}
For the DOS, compute the indices, Pearson correlation, Spearman correlation, Goodman-Kruskal \(\gamma\), Kendall's \(\tau_{b}\), Stuart's \(\tau_{c}\), Somers' D, lambda coefficients, and uncertainty
Many measures of association for two-way frequency tables consisting of two ordinal variables are based on the numbers of concordant and discordant pairs. To compute such indices, it is important to
Suppose \(x\) is a random variable with \(m\) levels such that \(\operatorname{Pr}(x=i)=p_{i}\) for \(i=\) \(1,2, \ldots, m\) with \(\sum_{i=1}^{m} p_{i}=1\). In other words, \(x \sim \mathrm{MN}(1,
Let \(x\) be a binary variable with outcomes 0 and 1 . Let \(p=\operatorname{Pr}(x=1)\). Show that entropy has the maximum at \(p=0.5\).
For an \(r \times s\) table, the probability of concordant (discordant) pair \(p_{s}\left(p_{d}\right) \leq \frac{m-1}{m}\), where \(m=\min (r, s)\).
EPDS is an instrument (questionnaire) for depression in postpartum women. This instrument is designed so that a person with a higher EPDS score has a higher chance to be depressed. Use the PPD data
Suppose \(x\) is a random variable with at least two levels, with \(\operatorname{Pr}\left(x=x_{i}\right)=p_{i}\), for \(i=1,2\). Let \(x^{\prime}\) be the new random variable based on \(x\) with the
If a fair die is thrown, then each number from 1 to 6 has the same chance of being the outcome. Let \(X\) be the random variable to indicate whether the outcome is 5,
For random variables \(X\) and \(Y\), show that \(E[E(X \mid Y)]=E(X)\) and \(\operatorname{Var}(X)=\operatorname{Var}(E(X \mid Y))+E(\operatorname{Var}(X \mid Y))\).
The sequence \(\left\{\frac{1}{n}\right\}_{n=1}^{\infty}\) converges to 0 . If we treat each constant, \(\frac{1}{n}\), as a constant random variable, then the corresponding \(\mathrm{CDF}\)
Suppose \(X_{n} \sim \chi_{n}^{2}\), the chi-square with \(n\) degrees of freedom. Show that\[\frac{1}{\sqrt{2 n}}\left(X_{n}-n\right) \rightarrow_{d} N(0,1)\]
Let \(X_{1}, \ldots, X_{n}\), be a sequence of i.i.d. random variables that follow the Poisson distribution with mean \(\mu\). Then the sample average \(\bar{X}_{n}=\frac{X_{1}+\cdots+X_{n}}{n}\) is
Prove Slutsky's theorem.
Prove that under some regularity conditions such as the exchangeability of the integral and differentiation, we have(a) \(E\left[\frac{1}{f\left(X_{i}, \boldsymbol{\theta}\right)}
A random variable \(X\) follows an exponential distribution with parameter \(\lambda\) that takes positive values and \(\operatorname{Pr}(X
For an independent sample of \(Y_{i}\) and \(X_{i}(1 \leq i \leq n)\), suppose that\[\begin{equation*}Y_{i}=\mathbf{X}_{i}^{\top} \beta+\epsilon_{i}, \quad \epsilon_{i} \sim\left(0,
Let \(\mathbf{f}(\boldsymbol{\theta})\) be a \(n \times 1\) and \(\mathbf{g}(\boldsymbol{\theta})\) a \(1 \times m\) vector-valued function of \(\boldsymbol{\theta}=\left(\theta_{1}, \ldots,
Use the properties of differentiation in Problem1.11 to prove (1.13) and (1.14).Problem1.11 To prove (1.13) and (1.14). 1.11 Let f (0) be an x1 and g (0) a 1 xm vector-valued function of 0 =
Prove (1.21). ZEEE = E- (XX) E [X; (Y; X] B) X] E (X;X). - (1.21) n
Let \(\mathbf{X}_{i}(1 \leq i \leq n)\) be an i.i.d. sample of random vectors and let \(\mathbf{h}\) be a vector-valued symmetric function \(m\) arguments.
Show that the U-statistic \(\widehat{\sigma}^{2}\) in (1.24) is the sample variance of \(\sigma^{2}\), i.e., \(\widehat{\sigma}^{2}\) can be expressed as \(\widehat{\sigma}^{2}=\frac{1}{n-1}
Consider the function \(\mathbf{h}\left(\mathbf{Z}_{1}, \mathbf{Z}_{2}\right)\) for the U-statistic in (1.22).(a)
Let \(\mathbf{X}_{i}(1 \leq i \leq n)\) and \(\mathbf{Y}_{i}(1 \leq i \leq n)\) be two independent i.i.d. samples of random vectors from two different population. Let \(\mathbf{h}\) be a
Install the statistical software packages that you will use for the book in your computer. Read the DOS data set using your statistical software and find out the number of observations and the number
Referring back to Problem 19, the same doctor then used the three variables—pain level (pain), age, and gender—to predict whether or not patients were admitted to the hospital 1admit = 12 by the
When the requirements of Pearson’s r cannot be met, we might still be able to employa. A t ratio.b. A parametric measure of correlation.c. A nonparametric measure of correlation.d. None of the above
Using SPSS, analyze the General Social Survey to find out if respondents’ attitudes toward assistance with healthcare costs for the sick (HELPSICK) are related to their attitudes about the
A researcher wonders whether people will give similar answers to questions about abortion depending upon the circumstances. In particular, she wants to know whether people will differ (or not) when
The phi and contingency coefficients are extensions of which test?a. Pearson’s rb. Spearman’s rank orderc. Chi-squared. Gamma
A researcher suspects that there may be a relationship between political views (POLVIEWS) and attitudes toward national support on issues related to foreign aid (NATAID), military, armaments and
To use a Spearman’s correlation coefficient, the data must bea. Measured at the interval level.b. Measured at the categorical level.c. Normally distributed.d. Rank-ordered with respect to the
The General Social Survey includes a variety of questions on activities that Americans may do at night. Apply Gamma to test the null hypothesis that there is no relationship between each of the
For Spearman’s correlation coefficient, if two (or more) positions are tied (assigned the same rank), you must adjust the data bya. Calculating the “difference” using the variance of the
Are males more likely than females to have ever injected drugs? Find phi for testing the null hypothesis that there is no relationship between ever injecting drugs (EVIDU) and sex (SEX).
To test the statistical significance of a Spearman’s correlation (rs) with a sample size larger than 30, we calculate aa. z statistic.b. t statistic.c. x2 statistic.d. F statistic.
Use the General Social Survey to find Cramér’s V to test the null hypothesis that opinions about whether marijuana should be made legal (GRASS) do not differ by marital status (MARITAL).
Goodman’s and Kruskal’s gamma is used whena. Samples sizes are small, and we have collected interval data.b. Sample sizes are small, and we have collected ordinal data.c. There are a large number
An expectant mother is looking for the best hospital possible in which to have her baby. She looks at a ranking of hospitals based on the total number of births delivered without complications. Being
A television network producer is concerned that sensitivity to victimization from sexual violence may negatively impact show ratings, so he ranks his network’s shows based on their number of
A social worker is interested in the link between how difficult and demanding a rehabilitation program is for hip replacements, and how satisfied the patient is when the program is completed. Using
A sociologist is interested in philanthropy and civil society, so she examines the relationship between amount of money donated and amount of time donated for a sample of seven wealthy donors. She is
A researcher is interested in how race affects the perception of racism in America, so he samples 120 individuals who are either black or white and asks them if they believe that racial
A graduate student interested in political sociology comes across research that theorizes that the presence of certain types of natural resources in a region may lead to different forms of political
Choose a variable from the General Social Survey to calculate a 95% confidence interval. Remember it has to be an interval level variable or a dichotomous (two category) variable
Analyze the General Social Survey to calculate a chi-square statistic to test the following hypothesis: votes for President in 2008 (Obama or McCain) will differ by sex (SEX).
Use the General Social Survey to calculate a chi-square statistic to test the following hypothesis: having a gun in the home (OWNGUN) will differ by whether you or your spouse hunts (HUNT).
For a chi-square value to be statistically significant, the following must be true:a. The median of one group must be nearly the same as the other groups.b. The mean of one group must be nearly the
Conference organizers wanted to make sure that there was about the same number of attendees at each event sponsored by a particular business school. Conduct a one-way chisquare analysis on the
Using the General Social Survey, calculate a chi-square statistic to test the null hypothesis that attitudes toward abortion for any reason (ABANY) do not vary by age (AGE).a. How large will the
Recode age into three groups 118 to 39 = 12140 to 59 = 22160 and above = 32 by using Transform then Recode into Different Variables and naming the new variable AGEGRP3 (for age grouped in three
A policy maker in a department of urban development is faced with budget cuts, and needs to select a public park to demolish. Faced with a hard decision, he determines the best way to choose is to
A committee of students wanted to know whether student activity fees were being charged fairly. To determine whether commuting and resident students differed in their use of campus activities, the
Analyze the General Social Survey to generate a single correlation matrix which will allow you to test the following null hypotheses:Create a correlation matrix of the three variables. Report the
Besides studying time, intelligence itself may be related to test performance. Suppose then that IQ (Z) is added to the two variables in Problem 21.Data From Problem 21In preparing for an
An education specialist is interested in how the number of words learned within the first two years of a child’s life affects the trajectory of academic performance later in high school. To test
Using the correlation coefficient results from Problem 27, determine the statistical significance of the relationship. Can the null hypothesis be rejected? In other words, does this relationship hold
The following is a correlation matrix among family size (X), weekly grocery bill (Y), and income (Z) for a random sample of 50 families.a. Which of the correlations are significant at the .05
When respondents who score high on the SATs tend also to get high grades in college, whereas those who score low on the SATs do poorly in college, there is reason to posita. A positive correlation
Generate a single correlation matrix which will allow you to test the null hypothesis of no relationship for all of the following pairs of variables from the General Social Survey:Job satisfaction
Use the General Social Survey to calculate a chi-square statistic to test the null hypothesis that opinions about whether marijuana should be made legal (GRASS) do not differ by marital status
Analyze the General Social Survey to calculate a chi-square statistic to test the following hypothesis: fear to walk at night in your own neighborhood (FEAR) will differ by sex (SEX).
Use the General Social Survey to calculate a chi-square statistic to test the following hypothesis: opinions about capital punishment (CAPPUN) will differ by sex (SEX).
Use the General Social Survey to calculate a chi-square statistic to test the following hypothesis: votes for President in 2008 and 2004 (PRES08 and PRES04) will be independent of the party
Does the rate of serious mental illness among substance abusers vary according to the type of substance abused? To find out, a psychiatrist collected data on the mental health of a random sample of
Analyze the General Social Survey to calculate a chi-square statistic to test the following hypothesis: opinions about the aged living with their children (AGED) will differ by sex (SEX). Recode age
A university president is curious about the likelihood that undergraduates would study for a semester in another country. She speculates that, given prevailing social norms supportive of males’
The chi-square test comparesa. The difference in means for two groups at a time.b. The difference in means for more than two groups.c. The variance of one group with the variance of others.d. The
The same social researcher as in Problem 38 then focuses on differences in livability by region. Selecting eight cities from each region of the country at random, he obtains the following rankings
A college president decides to examine how graduates from different majors fare in the job9 market. She collects the following starting salary data for a random sample of students in the Business
The Dean of the Business School at the same university as in the problem above then decides to examine how graduates from different majors within his school fare in the job market.Because the sample
The median test should be used toa. Determine the likelihood that a sample’s median is actually different from “0.”b. Determine the likelihood that two or more random samples have been taken
A major national magazine has ranked 200 American cities in terms of their livability. A social researcher selects 10 large cities and 10 small cities at random, and wishes to test whether there is a
Using the General Social Survey, calculate a chi-square statistic to test the null hypothesis that views about nuclear energy (NUKEELEC) do not vary by political party affiliation (PARTY).a. How
Using the General Social Survey, calculate Pearson’s r to test the following null hypotheses:a. Create the correlation matrix.b. Report the strength and direction of the Pearson’s r correlation
In the following graph, what type of correlation is depicted?a. Positive, strongb. Positive, moderatec. Negative, strongd. Negative, weak Text messages sent ce 30 25 20 15 10 5 0 0 5 10 15 Boredom at
In the following graph, what type of correlation is depicted?a. Positive, strongb. Positive, moderatec. Negative, strongd. Negative, moderate Text messages sent 30 25 20 15 10 5 0 0 5 10 15 Boredom
In the following graph, what type of correlation is depicted?a. Positive, strongb. Positive, weakc. Negative, strongd. Negative, weak Text messages
A correlation coefficient expresses in a single numbera. The strength of a correlation.b. The direction of a correlation.c. Both the strength and the direction of a correlation.d. None of the above
A researcher wonders if there is a correlation between (X) people’s opinions of bilingual education and (Y) their opinions about whether foreign-born citizens should be allowed to run for
Use the General Social Survey to generate a single correlation matrix to test the null hypothesis of no relationship for all of the following pairs of variables: Age your first child was born
Choose two variables from the General Social Survey so that you may generate and interpret a Pearson’s r correlation to test the null hypothesis of no correlation between the variables.
Interpret the following correlation statistics with respect to strength and direction:a. r = 0.951b. r = -0.232c. r = 0.096d. r = -0.738e. r = 0.415
The purpose of a partial correlation is to:a. Test for a significant relationshipb. Control a relationship for a third variablec. Make curvilinear relationship into a linear oned. Calculate a

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