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statistics
elementary statistics in social research

Questions and Answers of Elementary Statistics In Social Research

Fifty voters are randomly selected from each religious group in a state and surveyed about their preferred political party.Identify the sampling technique used, and discuss potential sources of bias
A government official surveys 150 students of a school in Shanghai to study the eating habits of school-going children in the city.Identify the sampling technique used, and discuss potential sources
Use the seventh row of Table 1 in Appendix B to generate 6 random numbers between 1 and 600.Identify the sampling technique used, and discuss potential sources of bias (if any). Explain.Table 1 in
A study of the dietary habits of 359,264 Korean adolescents was conducted to find a link between dietary habits and school performance. Identify the population and the sample in the study.
Determine whether each number describes a population parameter or a sample statistic. Explain your reasoning.(a) A survey of 1000 U.S. adults found that 52% think that the introduction of driverless
Determine whether the data are qualitative or quantitative. Explain.(a) A list of debit card personal identification numbers(b) The final scores on a video game
Determine the level of measurement of the data set. Explain your reasoning.(a) A list of badge numbers of police officers at a precinct(b) The horsepowers of racing car engines(c) The top 10 grossing
Determine whether the study is an observational study or an experiment. Explain.(a) Researchers conduct a study to determine whether body mass index (BMI) influences mortality. To conduct the study,
An experiment is performed to test the effects of a new drug on high blood pressure. The experimenter identifies 320 people ages 35–50 years old with high blood pressure for participation in the
Identify the sampling technique used in each study. Explain your reasoning.(a) A journalist asks people at a campground about air pollution.(b) For quality assurance, every tenth machine part is
Which technique used in Exercise 7 could lead to a biased study? Explain.Data from Exercises 7Identify the sampling technique used in each study. Explain your reasoning.A journalist asks people at a
A survey of 95 shopkeepers in a commercial complex with 550 shopkeepers.Determine whether the data set is a population or a sample. Explain your reasoning.
The height of each athlete participating in the Summer Olympics.Determine whether the data set is a population or a sample. Explain your reasoning.
The value of purchase by every sixth person entering a departmental store.Determine whether the data set is a population or a sample. Explain your reasoning.
The number of children in 25 households out of 75 households in a neighborhood.Determine whether the data set is a population or a sample. Explain your reasoning.
The ages at which all the presidents of a country were elected.Determine whether the data set is a population or a sample. Explain your reasoning.
The incomes of the top 10 taxpayers of a country.Determine whether the data set is a population or a sample. Explain your reasoning.
A survey of 1500 employees worldwide found that 62% of the respondents working in a foreign country settle there.Identify the population and the sample. Describe the sample data set.
A survey of 328 children in a city in Belgium found that 86% planned to visit their grandparents during the summer vacation.Identify the population and the sample. Describe the sample data set.
Of the 112.5 million blood donations collected globally, approximately 50% are collected from high-income countries.Identify the population and the sample. Describe the sample data set.
A survey of 1468 laptop users found that 81% preferred the use of mouse over touchpad.Identify the population and the sample. Describe the sample data set.
To gather information about the best mutual funds listed on a recognized stock exchange website, a researcher collects data about 134 of the 1000 mutual funds.Identify the population and the sample.
Forty out of a high school’s 500 students who took the midterm examination received a C grade.Determine whether the number describes a population parameter or a sample statistic. Explain your
Out of the 40 million casualties in the UK during World War II, two million were reported to be civilians.Determine whether the number describes a population parameter or a sample statistic. Explain
Employee records show that all the employees in an organization have received substantial increments over their joining salaries.Determine whether the number describes a population parameter or a
In a survey of 650 teachers, 16% reported that there have been instances of bullying in their class.Determine whether the number describes a population parameter or a sample statistic. Explain your
In a survey, 834 employees in the United States were asked whether they thought their jobs were highly stressful. Of the 834 respondents, 517 said yes. Identify the population and the sample.
Determine whether each number describes a population parameter or a sample statistic. Explain your reasoning.1. A survey of several hundred collegiate student-athletes in the United States found
For each study, identify the population and the sample. Then determine which part of the study represents the descriptive branch of statistics. What conclusions might be drawn from the study using
The table shows sports-related head injuries treated in U.S. emergency rooms during a recent five-year span for several sports. Which data are qualitative data and which are quantitative data?
For each data set, determine whether the data are at the nominal level or at the ordinal level. Explain your reasoning. 1. 2. Top five U.S. occupations with the most job growth (projected 2024) Movie
Two data sets are shown at the left. Which data set consists of data at the interval level? Which data set consists of data at the ratio level? Explain your reasoning. New York Yankees' World Series
Verify that the log-likelihood of model (7.7) is \(\sum_{i=1}^{k}\left[n_{i} \lambda-\exp (\lambda)\right]\).(a) Compute MLE of \(\lambda\).(b) Compute the Pearson chi-square statistic and compare it
Use the technique discussed in Section9.4 to develop a GEE approach for zeroinflated Poisson model for count responses in longitudinal studies.Section9.4: 9.4 Marginal Models for Longitudinal Data
For Problem 6.8,(a) Fit the corresponding negative binomial model with the same linear predictor.(b) Compare the analysis between part (a) and that from Problem 6.8. 6.8 For the Sexual Health pilot
Estimate the overall odds ratio of the set of tables in Problem3.6 and test whether the odds ratios are the same across the tables.Problem3.6 is: 3.6 Use the DOS data to test whether there is gender
Suppose \(\mathbf{x} \sim M N(n, \boldsymbol{\pi})\) follows a multinomial distribution of size \(n\) and probability \(\pi\). Derive the variance matrix of \(\mathbf{x}\).
Follow the steps below to prove the LLN without using CLT.(a) Let \(X\) be a random variable with mean \(\mu\) and variance \(\sigma^{2}\). Then for any real number \(\alpha>0,
(a) The longitudinal data set "v4c" is in the vertical format; transform it into the horizontal format.(b) Transform the data set you obtained in part (a) back into the vertical format.
Plot the mean/SD of HIV knowledge of adolescent girls at baseline and three months post treatment stratified by treatment for the Sexual Health study.
Perform some exploratory analysis on the DTS study described above.(a) Compute the mean and standard deviation of the HamD scores for the two treatment groups at each time point.(b) Treat repeated
For the DTS study, use subjects with all five assessments in HamD scores in the CAU group for this question. The intraclass correlation coefficient among the repeated measures in Ham-D scores can be
Suppose \(x \sim N\left(0, u^{2}\right), y \sim N\left(0, \sigma^{2}\right)\), and \(x\) is independent of \(y\). Given \(x+y=1\), find the MLE for \(x\).
In this question we develop a regression model to assess the treatment effect for stigma in the DTS study, controlling for demographics and baseline measurements. We will use the cumulative logit
Generalize the model considered in Example 4.11 to a marginal model for the longitudinal DOS data and compare the findings with that in Example4.11 Example 4.11 For the models in Example 4.8 DOS,
Prove (9.31) . = BE (GS;S; G) BT, B = (B), B-T (9.31)
Show that \(\mathbf{x}_{i} \perp y_{i t} \mid \mathbf{x}_{i t}\) and \(E\left(y_{i t} \mid \mathbf{x}_{i t}\right)=\mu_{i t}\) imply the FCCM.
Consider the GLMM in (9.11) with a logit link. Show that(a) \(E\left(y_{i t} \mid \mathbf{x}_{i t}, \mathbf{z}_{i t}\right) \approx \Phi\left(\frac{\mathbf{x}_{i t}^{\top}
Consider the GLMM in (9.11), with a log link. Show that(a) \(E\left(y_{i t} \mid \mathbf{x}_{i t}, \mathbf{z}_{i t}\right)=E\left[\left.\exp \left(\frac{1}{2} \mathbf{z}_{i t}^{\top} \Sigma_{b}
Construct a generalized linear mixed-effects model for the longitudinal DOS data with the fixed-effects component similar to that in Problem9.12 and a random intercept and assess the model
Construct a generalized linear mixed-effects model for the Sexual Health study data with the fixed-effects component similar to that in Example9.7 and random slopes of the time effect and assess the
Assess the models in Problems 9.12 and 9.14Problem 9.12 is:and Problem 9.14 is:. 9.12 Generalize the model considered in Example 4.11 to a marginal model for the longi- tudinal DOS data and compare
Use the longitudinal Catheter Self Management Study data for this question ("intake.csv" contains demographic and baseline information, and "catheter.csv" contains follow-up measurements). We model
In this question, we change the response in Problem9.23 to a three-level response by grouping the counts of UTIs into three levels, 0,1 , and \(\geq 2\). Fit the following models and test whether
Show that for a binary diagnostic test, \(\mathrm{AUC}=\frac{1}{2}\) (sensitivity + specificity).
Show that the ROC curve is invariant under monotone transformation of the test variable.
Verify (10.1) and (10.2). Se(c) = 1- $ (c-a) = and Sp(c) (c), cR, (10.1)
Let \(S\) be a curve in the two-dimensional \(x-y\) plane defined parametrically by \(x=F(t)\) and \(y=G(t)\), where \(F\) and \(G\) are smooth functions. Show that the slope of the tangent line at
Show that a binormal ROC curve is improper if the two normal distributions for diseased and nondiseased have different variances.
Show that for the binormal model in (10.3), if properness is further assumed, then this model reduces to the cumulative probit model with the disease status as the predictor and the ordinal test
Let \(t_{k}\) be the test outcome for the diseased \((k=1)\) and nondiseased \((k=0)\) subject. Show(a) \(\mathrm{AUC}=\operatorname{Pr}\left(t_{1} \geq t_{0}\right)\) if \(t_{k}\) is continuous;(b)
Express the AUC of a binormal ROC curve in terms of the means and variances of the two underlying normal distributions.
Show that the estimate in (10.5) equals the area under the empirical ROC curve. n1 no AUC = nino I{toj
In assessing the accuracy of HAM-D for the DOS, treat the SCID diagnosis of depression as a gold standard to(a) Estimate the ROC curve;(b) Estimate the AUC;(c) Which cut-points would you suggest
For the DOS, we would like to manually calculate two two-category NRIs when medical burden is added to HAM-D in the prediction of depression, usinga) 0.5 andb) the sample depression rate as the
Verify (10.7) and show(a) \(C_{b} \leq 1\);(b) \(ho_{C C C}=ho_{\text {Pearson }}\), if and only if \(\mu_{1}=\mu_{2}\) and \(\sigma_{1}=\sigma_{2}\).
For the Sexual Health pilot study, compute CCC and ICC between the diary and retrospective recall outcomes for the number of instances of unprotected vaginal sex.
For the domain sampling model described in Section10.4.1, show(a) \(p_{1}=\sqrt{\bar{ho}_{\infty}}\);(b) \(p_{K}=\sqrt{\frac{K \bar{ho}_{\infty}}{1+(K-1) \bar{ho}_{\infty}}}+o(1)\), where \(o(1)\) is
For the domain sampling model described in Section10.4.1, show(a) If \(\operatorname{Var}\left(y_{k}\right)=\sigma^{2}\) is a constant, the Spearman-Brown \(ho_{K}\) and Cronbach coefficient alpha
Show that CCC ranges between -1 and 1 and identify the scenarios in which CCC takes the value \(1,-1\), and or 0 .
Show that the moment-based estimate \(\widehat{ho}_{C C C}\) in (10.8) is consistent. pccc = 2812 s+s+ (1.2.) (10.8)
Let \(y_{i k}\) be a continuous outcome for the \(k\) th instrument from the \(i\) th subject \((1 \leq i \leq\) \(n, 1 \leq k \leq K)\). Assume that \(y_{i k}\) follows the LMM in (10.11). Let
Estimate the reliability index and the Cronbach coefficient alpha for each of the eight domains of CSF-36 based on the study described in Example10.4. Assess whether each item is coherently
Assess the test-retest reliability for each of the domains of CSF-36 based on the study described in Example10.5. Example 10.5 Consider the random subsample of 197 patients in the CSF-36 study who
Suppose for an i.i.d. sample of size \(n\), the disease status, \(d_{i}\), is MCAR with the probability of each \(d_{i}\) being observed given by \(\pi=0.75\).(a) Show that \(\frac{1}{n} \sum_{d_{i}
For the Sexual Health study, check whether the missingness of three-month posttreatment HIV knowledge is MCAR.
For Example 11.4, we are interested in the sensitivity and specificity of the test.(a) Compute the MLEs of sensitivity and specificity and their asymptotic variances based on the likelihood
Prove that the estimating equations in (11.13) are unbiased under MCAR, but are generally biased without the stringent MCAR assumption. (x) [y - f (xt;)] = 0, i=1 (11.13)
Show that the estimating equations (11.16) are unbiased. IM- G(xi) [Tiyi (1-ri) g (xi, Zi; Y) - f (xi; B)] = 0, (11.16) i=1
Prove that the estimating equations (11.20) are unbiased. 17 n n i i=1 G(xi) [yi- f (xi;)] = 0, (11.20)
Use mean score, IPW, and MI methods to estimate the sensitivity and specificity of the test in Example 11.1. Example 11.1 Suppose that we are interested in estimating the prevalence of a disease
Use the simulated DOS baseline data (with missing values in depression diagnosis) to assess the accuracy of HAM-D in diagnosis of depression.(a) Estimate the ROC curve using mean score, IPW, and MI
Show that the estimating equations in (11.35) are unbiased. Wn n n (B) = Gi (xi) AiSi = Gi (xi) Ai (yi - h) = 0, i=1 i=1 (11.35)
Prove (11.32) . E (Yi,k | Zi = 0, = e) = E (Yi,k | i = 1, = e) = E (Yi,k | Ti = e), k = 1,2. (11.32)
to the original three-level scale and compare the results from the two versions of the depression diagnosis variable.
For a simple random sample from a population of size \(N\), the subjects are not sampled independently because of the finite size of the population.(a) Show that the probability of being sampled for
Compute the variance of \(\widehat{\mu}\) in (11.45) via the following steps.(a) Suppose the \(N\) subjects of the population are labeled from 1 to \(N\). The \(i\) th subject with outcome \(y_{i}\)
In a study to determine the distribution of time to the occurrence of cancer after exposure to certain type of carcinogen, a group of mice is injected with the carcinogen, and then sacrificed and
Under the race set up in Section8.1.2, think a scenario when you may have left truncation issue. 8.1.2 Truncation Another issue arising in the analysis of time to event data is truncation. Under
Given that a subject survives up to and including time \(t\), how likely is the failure to occur within the next infinitesimal time interval \((t, t+\Delta t)\) ?
For a continuously differentiable survival function \(S(t)\), the hazard function is defined as \(h(t)=-\frac{S^{\prime}(t)}{S(t)}\). Prove that \(S(t)=\exp \left(-\int_{0}^{t} h(s) d s\right)\).
For \(T \sim\) exponential \((\lambda)\), the conditional distribution of \(T-t_{0}\), given \(T \geq t_{0}\), again follows an exponential \((\lambda)\) distribution.
Plot the survival and hazard function for the exponential and Weibull survival times using different parameters and check their shapes.
Let \(\left\{\pi_{j}\right\}_{j=1}^{k-1},\left\{S_{j}\right\}_{j=1}^{k-1}\), and \(\left\{p_{j}\right\}_{j=1}^{k-1}\) be defined in (8.1), (8.2), and (8.3). Show that any one of them determines the
Derive the likelihood (8.5) based on (8.4). L= n Ip(ti, Br)] [1p (tj, r)] S (tj1, r). i=1 (8.5)
Prove that (8.7) provides the ML estimates based on the likelihood (8.5) and (8.6). n \L = [ [ [p (tj, )]^* [1 p (tj, r)]ci S (tj1, T). (8.5) i=1
For the DOS, we are interested in the time to drop out of the study.(a) Create a life table including the number of subjects at risk, the number of failures (new depression), the number of survivors,
Verify the likelihood (8.16). n t-1 II (Pt, (xi) i (1 Pt, (x;))(1-cs) II (1 Pk (xi)) - (8.16) i=1 k=1
Let \(h_{0}(t)\) denote the hazard function when \(\mathbf{x}_{i}=\mathbf{0}\), and \(S_{0}(t)=\exp \left(\int_{0}^{t} h_{0}(u) d u\right)\) be the corresponding survival function. If \(h\left(t,
Use the delta method to prove (8.9). (tj) l=1 1-Pi nipi 1/2 (8.9)

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