# Significance Tests Statistics and Probability Questions & Answers

Recent questions in Significance tests
Ernesto Wagner 2022-11-23

### Perform hypothesis test for population mean It is claimed that average rice production by Company2 is 209,500 cwt (Company2 = 209500.0). Test this claim using a hypothesis test at 1% level of significance.My answer: Null = 209500.0 alternative = .01I'm being told that these my answers are not correct. I have no idea why. Any help would be appreciated!

Abdiel Mays 2022-11-13

### Suppose ${X}_{n}$ is an i.i.d. random sample from the $N\left(\mu ,{\sigma }^{2}\right)$ population, where μ is unknown but ${\sigma }^{2}$ is known. Consider a test statistic $T=\sqrt{n}{\left(}^{\prime }bar{X}_{n}-{\mu }_{0}\right)/\sigma$ at the significance level $\alpha$ for ${H}_{0}:\mu ={\mu }_{0}\phantom{\rule{1em}{0ex}}versus\phantom{\rule{1em}{0ex}}{H}_{a}:\mu \ne {\mu }_{0}$. Find Type I error of this test.Confusion:One of my friends gives this answer:The hypothesis model will be${H}_{o}:\mu ={\mu }_{o}$${H}_{A}:\mu \ne {\mu }_{o}$Type I error $\alpha$ is the probablity of rejecting the null hypothesis ${H}_{o}$Thus $p=\alpha$I think it's not correct, because Type I error is the the probablity of rejecting the null hypothesis ${H}_{o}$ when ${H}_{o}$ is true , and α means the size of test is no larger than $\alpha$, so I think there are some differences between these two concepts.So who is wrong? why? and what the correct answer should be?

reevelingw97 2022-11-11

### If you are given a population of students doing hypothesis tests for a certain condition at a certain significance level, is it possible to calculate:(a) how many students will fail to reject the null hypothesis given that the null hypothesis is false(b) how many students will reject the null hypothesis given that the null hypothesis is true.I have tried searching online but so far all sites only show how to calculate the probability that at least one type I error will be made. Any assistance will be greatly appreciated.

Audrey Arnold 2022-11-11

### Suppose ${X}_{n}$ is an i.i.d. random sample from the $N\left(\mu ,{\sigma }^{2}\right)$ population, where μ is unknown but ${\sigma }^{2}$ is known. Consider a test statistic $T=\sqrt{n}{\left(}^{\prime }bar{X}_{n}-{\mu }_{0}\right)/\sigma$ at the significance level $\alpha$ for ${H}_{0}:\mu ={\mu }_{0}\phantom{\rule{1em}{0ex}}versus\phantom{\rule{1em}{0ex}}{H}_{a}:\mu \ne {\mu }_{0}$. Find Type I error of this test.Confusion:One of my friends gives this answer:The hypothesis model will be${H}_{o}:\mu ={\mu }_{o}$${H}_{A}:\mu \ne {\mu }_{o}$Type I error $\alpha$ is the probablity of rejecting the null hypothesis ${H}_{o}$Thus $p=\alpha$I think it's not correct, because Type I error is the the probablity of rejecting the null hypothesis ${H}_{o}$ when ${H}_{o}$ is true , and α means the size of test is no larger than $\alpha$, so I think there are some differences between these two concepts.So who is wrong? why? and what the correct answer should be?

Jacoby Erickson 2022-10-25

### You have a coin and p is the probability of heads.You throw the coin until you obtain heads for the first time.You want to test ${H}_{0}:p=1/2$ against ${H}_{1}:p<1/2$. You reject ${H}_{0}$ in favor of ${H}_{1}$ if $T\ge k$, where T is the number of the throw which yielded heads for the first time and k an integer.Determine the smallest value of k corresponding to the level of significance α=0.01.Answer k=8I can't find the correct solution for this question, how it's possible to have ${H}_{1}:p<1/2$ and then they say that we reject ${H}_{0}$ in favor of ${H}_{1}$ if $T\ge k$, shouldn't they be in the same direction?I have to compute $P\left(T>k|{H}_{0}\right)=P\left(T>k|p=1/2\right)$ right? How can I compute it?

Emilio Calhoun 2022-10-24

### A chi square test is conducted to check whether a person's ability in Mathematics has an impact on his/her interest in Statistic. The test statistic is 13.277 under the tested null hypothesis. write a recommended null hypothesis and an alternative hypothesis. Briefly describe your conclusion on this test at the 0.01 significance level.

erwachsenc6 2022-10-23

### check the null hypothesis that the mean is extra than or identical to 50. Use a one tail and a importance level of .05. A sample of 25 answers has an average=forty five and a variance=25.

Lilah Hurst 2022-10-21

### Given there's a number generator which creates irregular numbers to 9. Given after 50 number eras as it were 5 of the created numbers are underneath 3 (i.e. 0,1 or 2) and the importance level is α = 1%. How can I test in the event that for this importance level the number generator is producing all numbers with equal probability? It is obvious to me that in the event that the number generator would be without a doubt producing the numbers with uniform likelihood, one would anticipate 15 created numbers between and 2 - but how do I test the nullhypothesis in this case accurately?

Winston Todd 2022-10-18

### We have the Random Variable X, which is $\mathrm{\Gamma }\left(p,\lambda \right)$ distributed with the density:${f}_{p,\lambda }\left(x\right)=\frac{{\lambda }^{p}}{\mathrm{\Gamma }\left(p\right)}\cdot {x}^{p-1}\cdot {e}^{-\lambda x}$with p=10 and ${H}_{0}:\lambda =2$ or ${H}_{1}:\lambda =4$ and $\alpha =0.001$I want to apply the Lemma of Neyman Pearson which states:Be c>0 fixed and chosen in the way that $A\left(c\right)=\left\{x\in B:\frac{{f}_{0}\left(x\right)}{{f}_{1}\left(x\right)}\ge c\right\}$ such that ${\mathbb{P}}_{{H}_{0}}\left(X\in A\left(c\right)\right)=\alpha$Then the test with the region A(c) among all tests with significance level $\alpha$ is the most powerful.I am now trying to calculate A(c), but got stuck. I have:${\int }_{A\left(c\right)}{f}_{0}\left(x\right)dx={\int }_{A\left(c\right)}\frac{{\lambda }^{p}}{\mathrm{\Gamma }\left(p\right)}\cdot {x}^{p-1}\cdot {e}^{-\lambda x}dx=\alpha .$But I don't know how to get A(c) from this integral...$\frac{{f}_{0}\left(x\right)}{{f}_{1}\left(x\right)}=\frac{1}{1024}\cdot {e}^{2x}$

Kasey Reese 2022-10-17

### Is P value Type I error in hypothesis testing?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?

Tara Mayer 2022-10-16

### Why "bother" with a null hypothesis at all?whenever i am seeking to get into data (once more), i am constantly lost at speculation trying out.My simple question is - why can we form a null hypothesis as a negation of what we need to show within the first region, and most effective then can we show or disprove the null speculation?Why do we do it at all, in preference to just proving the authentic speculation?

Kevin Charles 2022-10-11

### A poll from a previous year showed that 10% of smartphone owners relied on their data plan as their primary form of internet access. Researchers were curious if that had changed, so they tested ${H}_{0}:p=10\mathrm{%}$ versus ${H}_{a}:p\ne 10\mathrm{%}$ where p is the proportion of smartphone owners who rely on their data plan as their primary form of internet access. They surveyed a random sample of 500 smartphone owners and found that 13% of them relied on their data plan.The test statistic for these results was $z\approx 2.236$, and the corresponding P-value was approximately 0.025.Assuming the conditions for inference were met, which of these is an appropriate conclusion?a) At the $\alpha$=0.01 significance level, they should conclude that the proportion has changed from 10%.b) At the $\alpha$=0.01 significance level, they should conclude that the proportion is still 10%.c) At the $\alpha$=0.05 significance level, they should conclude that the proportion has changed from 10%.d) At the $\alpha$=0.05 significance level, they should conclude that the proportion is still 10%.The correct answer is c but why could it not have been b? Why is it c?

dalllc 2022-10-09

### 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.Solution:${H}_{0}$: mean = 120 (null hypothesis) ${H}_{1}$: mean > 100 (alternative hypothesis)we will use z test as the sample count is more than 30$z=\frac{120-100}{60\sqrt{64}z}=2.67$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 100

Jamarcus Lindsey 2022-10-08

### 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.Solution:${H}_{0}$: mean = 120 (null hypothesis) ${H}_{1}$: mean > 100 (alternative hypothesis)we will use z test as the sample count is more than 30$z=\frac{120-100}{60}\sqrt{64}z=2.67$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 100

kasibug1v 2022-09-09

### Equations of significance probabilitiesConsider a population of independent light bulbs with an exponential lifetime distribution with mean $\mu$. It is claimed that their expected lifetime is 1000 hours. A definition of a 100(1−$\alpha$)% confidence interval obtained from an observation to is the set of all ${\mu }_{0}$ which are not rejected in a test of a null hypothesis ${\mu }_{0}$ against an alternative hypothesis $\ne$.One particular light bulb fails after 622 hours. Solve the equations of the two significance probabilities Pr(T "" 622 |${\mu }_{0}$) = 0.05 (for a test of ${\mu }_{0}$ versus ${\mu }_{0}$) and Pr(T "" 622 |${\mu }_{0}$ = 0.05 (for a test of ${\mu }_{0}$ versus ${\mu }_{0}$) for $\mu$. Determine the range of values of $\mu$ such that both of the probabilities Pr(T "" 622 | $\mu$) and Pr(T "" 622 |$\mu$) are at least 0.05. (This range gives an equi-tailed 90% confidence interval for $\mu$.)I don't seem to understand what they mean by 'solve the equations'. Do I have to find a specific value for T or compute Pr(T $\ge$ 622 |${\mu }_{0}$), Pr(T $\le$ 622 |${\mu }_{0}$) and compare with 0.05? I believe I will get the second part after I understand this bit.

gsragator9 2022-09-01

### Two Tail Hypothesis Test with Variancetest the hypothesis the variance for economists equal the variance for the historians. Use a .05 significance level, a two tail test and the following data :economist historianvar 120 90n 46 38

ntsibengshete81 2022-08-25

### A local newspaper claims that 90% of its online readers are under the age

gorgeousgen9487 2022-07-14

### Here's a problem I thought of that I don't know how to approach:You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with significance level $\alpha$, where your null hypothesis is that the coin is fair and your alternative hypothesis is that the coin is not fair. In terms of $\alpha$, what is the expected number of flips before the first time that you reject the null hypothesis?Edit based on comment below: For what values of α is the answer to the question above finite? For those values for which it is infinite, what is the probability that the null hypothesis will ever be rejected, in terms of $\alpha$?Edit 2: My post was edited to say "You believe that you have a fair coin." The coin is in fact fair, and you know that. You do the hypothesis tests anyway. Otherwise the problem is unapproachable because you don't know the probability that any particular toss will come up a certain way.

Wisniewool 2022-07-14

### Question: A researcher believes that the stock market performance and the property market performance in Singapore are associated. Describe one (1) statistical approach the researcher would implement to determine the association between the performances of these two markets. Explain their association and the factors that affect their association.

Almost all significance test practice problems that you will encounter below help to find solutions to your questions as the answers deal with the same equations that have been used. Start with any significance test example and you will understand that you only have to change variables to determine each value. It's exactly what makes significance test equation so popular as it provides help with more advanced probability concepts. As you look through significance test questions, look for similar patterns as these are where you must start regardless of whether you deal with a complex engineering project or statistical analysis.