 # College Statistics & Probability Questions and Answers

Recent questions in College Statistics affizupdaftf3opg 2022-05-07 Answered

### How can a median be greater than the mean? Brooklynn Hubbard 2022-05-07 Answered

### I'm confused about the interpretation of P value in hypothesis testing. I know that we set significance level as 0.05 which is the threshold we set for this test so that it won't suffer from Type I error by 5%.And we are comparing P to significance level, does it mean P is the probability of making type I error based on the sample? rynosluv101wopds 2022-05-07 Answered

### A sample is used both to construct a 95% confidence interval for a population proportion p and to run a significance test with null hypothesis ${H}_{0}:P=0.07$ and significance level $\alpha =0.05$.Is it possible that $p=0.7$ falls outside of the 95% confidence itnerval, yet ${H}_{0}$ is not rejected?I'm assuming that since a two-sided test at significance level α gives roughly the same result as a $100\left(1-\alpha \right)\mathrm{%}$confidence interval, this is not possible.Also, in terms of the sample size required for a significance test, does it just have to statisfy the normal condition - $n{p}_{0}\ge 10$and $n\left(1-{p}_{0}\right)\ge 10$? Edith Mayer 2022-05-07 Answered

### I was working on the following problem:Consider two probability density functions on $\left[0,1\right]:{f}_{0}\left(x\right)=1$ and ${f}_{1}\left(x\right)=2x$. Among all tests of the null hypothesis ${H}_{0}:X\sim {f}_{0}\left(x\right)$ versus the alternative $X\sim {f}_{1}\left(x\right)$, with significance level $\alpha =0.1$, how large can the power possibly be?I think we need to begin by looking at an arbitrary test with significance level$\alpha =0.1$ but I am having a hard time doing this. I am not even sure this is the direction we want to head in so I was hoping to get some hints regarding this problem. zuzogiecwu 2022-05-07 Answered

### I'm trying to solve following type of combined date given question. Imagined that there's two factories. They are X and Y. "X" factory has 10 workers and "Y" has 20 workers. These two factory's production respectively given as Σx^2=2950 and ΣY^2=5000. Moreover, "X" factory mean is 18 and "Y" factory mean is 15.If the whole population is considered as a one. How to calculate the mean and standard deviation in a situation like this one? I'm not expecting the final answer, I rather would like to know the procedure that how to solve such a question. Azzalictpdv 2022-05-07 Answered

### Why do electrons in a current spread apart, from their point of view?There are many videos explaining how special relativity causes magnetism, with the most relevant part being that from the reference frame of the electrons, the positively charged ions in the wire have gotten closer together, and the negatively charged electrons have spread apart, a non-zero electric charge in a wire. We can determine that this must happen using the fact that in a lab reference frame, the wire is electrically neutral, and applying a Lorentz transformation to the electrons and ions. However, the electrons must see a different mechanism causing the space between them to increase. They need to see something pushing them apart. They don't know that the wire needs be electrically neutral in another reference frame. What's this mechanism that they see? Does it have anything to do with the period of time when they're accelerating?I saw a bunch of questions that seemed similar to this one, but I didn't find any of the answers to be satisfactory. They all seemed to just explain how we know that the wire must be charged in different reference frames, using the Lorentz tranformation. Deven Livingston 2022-05-03 Answered

### Sampling Distribution of the Sample MeanConcerning estimations/intervals:True or False: if the three conditions Random, Normal, and Independent for using a confidence interval for a population mean are not met, then the sampling distribution of $\stackrel{―}{x}$ is unknown.I'm unsure about the criteria - would you not know the distribution of the sample mean even if these criteria are unfilled? Are there specific conditions for doing so? Deven Livingston 2022-05-03 Answered

### Random-digit-dialing telephone surveys used to exclude cell phone numbers. If the opinions of people who have only cell phones differed from those of people who have landline service, the poll results may not represent the entire adult population. The Pew Research Center interviewed separate random samples of cell-only and landline telephone users who were less than 30 years old and asked them to describe their political party affiliation.$\text{□}$ or goodness of fit, homogeneity or independence? NepanitaNesg3a 2022-05-02 Answered

### Show the difference between two approaches to significance testing? Paula Boyer 2022-05-01 Answered

### The latest worldwide virus has an infection rate of $0.1\mathrm{%}$ (that is, $1$ in $1000$ people actually have the virus). The chance that someone who has the virus tests positive is said to be $99\mathrm{%}$. The chance that someone who does not have the virus tests negative is also said to be $99\mathrm{%}$. What are the chances that someone who tests positive does not in fact have the virus (a “false positive”)? compyac 2022-05-01 Answered

### Show that the significant value of t at level of significance a for one-tailcd test is cqual to that of t at 2$\alpha$ significance level for two-tailed test. eszkortwxp 2022-05-01 Answered

### Sample mean to population meanHow to use sample mean to know the population mean? Now I have sample mean, sd and population mean. They are 37.28, 25 and 34. Sample size is 25 Here is my thought $N<25$. Sample mean may not equal to population mean. population mean is $37.28+\frac{25}{\sqrt{25}}=42$ Am i correct? dolovatgyp 2022-05-01 Answered

### Resources O Hint Press Esc to exit full screen Are younger people more likely to be vegan/vegetarian? To investigate, the Pew Research Center classified a random sample of 1480 U.S. adults according to their age group and whether or not they are vegan/vegetarian.Determine which chi-square test is appropriate for the given setting. Which response below gives the correct test with appropriate reasoning?a. Chi-square goodness of fit test because the data came from a single random sample with the individuals classified by their chocolate consumption.b. Chi-square test for homogeneity because the data come from two independent random samples – those who are vegan/vegetarian and those who are not.c. Chi-square test for homogeneity because the data came from a single random sample with the individuals classified according to two categorical variables.d. Chi-square test for independence because the data came from a single random sample with the individuals classified according to two categorical variables.e. Chi-square test for independence because the data come from two independent random samples – those who are vegan/vegetarian and those who are not. Paula Boyer 2022-05-01 Answered

### Let be a random sample from a normal distribution with known mean $\mu$ and unknown variance ${\sigma }^{2}$. Three possible confidence intervals for ${\sigma }^{2}$ area) b) c) where are constants.Find values of these six constants which give confidence level 0.90 for each of the three intervals when $n=10$ and compare the expected widths of the tree intervels in this caseWith ${\sigma }^{2}=1$, what value of n is required to achieve a $90\mathrm{%}$ confidence interval of expected width less than 1 in cases (b) and (c) above? Lymnmeatlypamgfm 2022-05-01 Answered

### Let $f:\mathrm{\Omega }⇒\mathbb{R}$ be a Borel-measurable function, X a random variable with values in $\mathrm{\Omega }$ and ${X}_{i}\in \mathbb{R},i\in \mathbb{N}$ realizations of X.LiteratureIn their book on Monte Carlo Methods (Simulation and the Monte Carlo method, Second Edition) Rubinstein and Kroese give in Section 4.2.1 an approximate confidence interval for the mean E[f(X)]:$\left(\stackrel{^}{\mu }±{z}_{1-\frac{\alpha }{2}}\frac{\stackrel{^}{{\sigma }_{f}}}{\sqrt{N}}\right),$where we write N for the number of Monte Carlo samples, ${z}_{1-\frac{\alpha }{2}}$ for the $1-\frac{\alpha }{2}$ quantile of the standard normal distribution, $\stackrel{^}{\mu }=\frac{1}{N}\sum _{i=1}^{N}f\left({X}_{i}\right)$ for the unbiased estimator of the mean and ${\stackrel{^}{\sigma }}_{f}=\sqrt{\frac{1}{N-1}\sum _{i=1}^{N}{\left(f\left({X}_{i}\right)-\stackrel{^}{\mu }\right)}^{2}}$ for the empirical standard deviation derived from the unbiased estimator of the variance.Extension to varianceIn my opinion we can derive in a similar way a confidence interval for the variance var[f(X)] if we define $g\left(X\right)\phantom{\rule{0.222em}{0ex}}={\left(f\left(X\right)-\stackrel{^}{\mu }\right)}^{2}$ and repeat the construction. This leads to the interval$\left({\stackrel{^}{\sigma }}_{f}^{2}±{z}_{1-\frac{\alpha }{2}}\frac{{\stackrel{^}{\sigma }}_{g}}{\sqrt{N}}\right),$where ${\stackrel{^}{\sigma }}_{g}=\frac{1}{N-1}\sum _{i=1}^{N}{\left(g\left(X\right)-\frac{1}{N}\sum _{i=1}^{N}g\left({X}_{i}\right)\right)}^{2}$How can I transfer this to give a confidence interval for the standard deviation? Surely I can't just take the square root of the boundaries. Any literature, help and thoughts are welcome. Thanks. bacfrancaiso0j 2022-04-30 Answered

### 3% of the population has disease X.A laboratory blood test has(a) 96% effective at detecting disease X, given that the person actually has it.(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.What is the probability a person has the disease given that the test result is positive? kabutjv7 2022-04-30 Answered

### Rewrite the probability statement$P\left(0 Wyatt Flores 2022-04-30 Answered

### Let X be a Poisson ($\theta$) random variable. Unbiased estimatorShow that ${\left(-1\right)}^{X}$ is an unbiased estimator for ${e}^{-2\theta }$ This is a fairly bad estimator for a number of reasons - so this exercise helps show why unbiasedness is not the most important criterion for an estimator. Treboldiqtw 2022-04-30 Answered

### Let ${X}_{1},\cdots ,{X}_{n}$ random sample of $U\left[\theta -\frac{1}{2};\theta +\frac{1}{2}\right]$.Consider a confidence interval for $\theta$. Find their confidence level and show that result is valid for any distribution symmetric around $\theta$ compyac 2022-04-30 Answered

### Let ${p}_{1}$= population proportion for population 1, ${p}_{2}$= population proportion for population 2, ... and ${p}_{k}$= population proportion for population k. Consider the following null hypothesis: ${H}_{0}$: ${p}_{1}$=${p}_{2}$=...=${p}_{k}$. Which of the following statements is correct?a. The alternative hypothesis to the null hypothesis stated above must be: ${H}_{\alpha }$: Not all population proportions are equal.b. If the sample data and the chi-square test computations indicate ${H}_{0}$ cannot be rejected, we cannot detect a difference among the k population proportions.c. If the sample data and the chi-square test computations indicate ${H}_{0}$ can be rejected, we have the statistical evidence to conclude that one or more population proportions differ from the other population proportions.d. All of the above.

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