 # Statistics & Probability Questions and Answers

Recent questions in Statistics and Probability Kaila Branch 2022-09-27

### What is the mode of the following numbers? $12,11,14,10,8,13,11,9$ $a11\phantom{\rule{0ex}{0ex}}b.10\phantom{\rule{0ex}{0ex}}c.14\phantom{\rule{0ex}{0ex}}d.8$ misyjny76 2022-09-26

### How do you find the exact value of $\mathrm{cos}2x$ given $\mathrm{cos}x=-\frac{2}{13},\frac{\pi }{2} ? Dymnembalmese2n 2022-09-26

### A survey of 64 medical labs revealed that the mean price charged for a certain test was Rs. 120, with a standard deviation of 60. Test whether the data indicates that the mean price of this test is more than Rs. 100 at 5% level of significance.I have solved this question but I don't know whether the answer is correct or not.H0: mean = 120 (null hypothesis) H1: mean > 100 (alternative hypothesis)we will use z test as the sample count is more than 30$z=|120-100|/60/\sqrt{64}\phantom{\rule{0ex}{0ex}}z=2.67$at 5% of significance, the critical value of z is 1.96. Since the z value we obtained is more than 1.96, so we reject the null hypothesis and therefore the mean price of the test is more than 100Please tell whether the answer is correct or there is some mistake in this. Help is appreciated. unjulpild9b 2022-09-26

### Whether to use $Z$ or $T$ test statistic for a sample of $50$, known sample variance and unknown population variance Nathanael Perkins 2022-09-26

### I have just started a course in statistics and have some general questions that have arisen trying to solve the following question:A survey organisation wants to take a simple random sample in order to estimate the percentage of people who have seen a certain programme. The sample is to be as small as possible. The estimate is specified to be within 1 percentage point of the true value; i.e., the width of the interval centered on the sample proportion who watched the programme should be 1%. The population from which the sample is to be taken is very large. Past experience suggests the population percentage to be in the range 20% to 40%. What size sample should be taken?I think I have to use this and solve for n$1.96\sqrt{\frac{\pi \left(1-\pi \right)}{n}}=.01$where $\pi$ is the sample estimated proportion of people who watch the programme.Now does this mean that I am 95% sure that I am within 1% accuracy? I also am not aware as to how I can find π though I have read I could use the population standard deviation instead and suspect I would have to use that as I am given some information- that the pop proportion is 20%-40%.Finally in general what is being said here:$\pi ±1.96\sqrt{\frac{\pi \left(1-\pi \right)}{n}}=.01$My notes at the moment just say it contains the population mean 95% of the time.... why? I think if I had some graphical understanding of what was going on everything would be much simpler for me. fion74185296322 2022-09-26

### How are the variance and the standard deviation of a distribution related? What is measublack by the standard deviation? imchasou 2022-09-26

### What is the statistics processes stages and how do we select a simple random ? Haven Kerr 2022-09-25

### Figuring out probability of two random events both happening So here's the problem:The table below shows the distribution of education level attained by US residents based on data collected during the 2010 American Community Survey:Highest level of education %Less than 9th Grade 0.109th to 12th no diploma 0.09High school grad - GED 0.25Some college No degree 0.23Associate's degree 0.08Bachelor's degree --Graduate or professional degree 0.09Answer the following questions (give all answers to 2 decimal places):a) Fill in the empty box for the proportion of US residents whose highest education level attained was a bachelors degree.b) If two individuals are chosen at random from the population, what is the probability that both will have at least a bachelors degree?c) If two individuals are chosen at random from the population, what is the probability that at least one will have some college or a college degree of some sort?d) If two individuals are chosen at random from the population, what is the probability that exactly one will have some college or a college degree of some sort?I managed to figure out a) which was 0.16. However, I tried a different methods to get b) but none of them worked. I answered .24, .25, and .01 but none of them were correct.To answer b) I used the following formula:P(A or B) = P(A) + P(B) - P(A & B) = (.16) + (.09) - (.16)(.09)which gave me .24 but that was incorrect. What am I doing wrong? Quinlan7g 2022-09-25

### The joint probability mass function of the random variables X, Y, Z is p$\left(1,2,3\right)=p\left(2,1,1\right)=p\left(2,2,1\right)=p\left(2,3,2\right)=\frac{1}{4}$ Find E[XY+XZ+YZ]. Medenovgj 2022-09-25

### Let X and Y be independent random variables each having a geometric density with parameter p. Find E[Y | X + Y = z] where z is a nonnegative integer. Kallie Fritz 2022-09-25

### I had this question when I was shopping for some rates to refinance my house: how do know if I have found the lowest rates after searching through a bunch of offers with a reasonable amount of confidence, in a mathematical/statistical term. For example, if found "x" offers from different vendors, is the amount of "x" large enough to represent the bigger population i.e. all offers out there?The reason I am asking is that I vaguely remember there are statistical sampling techniques from college days that survey companies used to represent the bigger populations. For instance, after surveying 2000 people, they can conclude something for the U.S. population. In other words, they "know" 2000 people can represent the U.S. population with some scientific backing.Just curious, if there is a way to figure out the magical number "x", then I can go with the lowest rate after browsing through "x" vendors as well. The question is purely theoretical in this simple case: the lowest rate wins. Any leads or ideas are appreciated. memLosycecyjz 2022-09-25

### Let ${Y}_{1},{Y}_{2},...,{Y}_{n}$ be independent, uniformly distributed random variables on the interval $\left[0,\theta \right]$. density function of ${Y}_{\left(n\right)}$ Camila Brandt 2022-09-25

### Confidence Intervals - Doubts on InterpretationsSuppose we use a sample mean $\overline{X}$ to construct a 95% confidence interval [a,b].I was told that it is incorrect to say that there is a 95% probability that the population mean lies between a and b. Because the population mean is a constant and not a random variable. The probability that a constant falls within any given range is either 0 or 1.However, from the textbook it is said that we expect 95% of the confidence intervals to include the population mean.If 95% of the confidence intervals are expected to include the population mean, then each confidence interval has 95% probability to include the population mean. Therefore I think it is correct to say there is a 95% probability that the population mean lies between a and b.Where did I make mistakes? Alexus Deleon 2022-09-25

### Intro to probability chapter 4 ex 31A group of 50 people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.Solution: Creating an indicator r.v. for each pair of people, we have that the expected number of pairs of people with the same birthday is (50C2 . 1/365) by linearity. Now create an indicator r.v. for each day of the year, taking the value 1 if at least two of the people were born that day (and 0 otherwise). Then the expected number of days on which at least two people were born is$365\left(1-\left(364/365{\right)}^{50}-50\cdot \left(1/365\right)\cdot \left(364/365{\right)}^{49}\right)$Could someone explain how we got the answer ? gaby131o 2022-09-25

### The $t$-statistic for the sample mean is$t=\frac{\overline{x}-\mu }{s}$How is the sample estimate of the standard error computed in this context? In particular, does it make use of the population mean or the sample mean? videosfapaturqz 2022-09-25

### We want to conduct a hypothesis test of the claim that the population mean score on a nationwide examination in anthropology is different from 492. So, we choose a random sample of exam scores. The sample has a mean of 505 and a standard deviation of 75. For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. Then calculate that statistic. Round your answers to two decimal places. The sample has size 12, and it is from a normally distributed population with an unknown standard deviation. Oz = = 0 O It is unclear which test statistic to use. Altenbraknz 2022-09-25

### Determine what type of sampling is used (Simple Random Sampling, Stratified Random Sampling, Systematic Sampling, Cluster Sampling, or Convenience Sampling). 1. To get a sense of election outcomes, a political group chooses ten precincts to conduct a survey of voters in those areas. waldo7852p 2022-09-25

### "... This means that if we used the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter to fall within the interval estimates 95% of the time. "However, to my knowledge, isn't this one of the common misconceptions of a confidence interval? And that the actual interpretation is that if we gather n amount of these confidence intervals, there is a 95% probability that the n collected intervals all contain the true parameter?My notes give this definition:Let $L:=L\left({X}_{1},\dots ,{X}_{n}\right)$ and $U:=U\left({X}_{1},\dots ,{X}_{n}\right)$ be such that for all $\theta \in \mathrm{\Theta }$,$\mathbb{P}\left(L<\theta \le U\right)\ge 1-\alpha$and then it says, that this is the probability $1-\alpha$ that the true parameter lies within this interval . Which one is correct? lunja55 2022-09-25
### Suppose that random variables X and Y vary in accordance with the joint pdf, ${f}_{X,Y}\left(x,y\right)=c\left(x+y\right),0. Find c. Alexus Deleon 2022-09-25