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Conduct a hypothesis test:For the past several years the mean number of people in household has been declining-A social scientist believes that in certain large &#x...

Question

Conduct a hypothesis test:For the past several years the mean number of people in household has been declining-A social scientist believes that in certain large ' city; the mean number of people per household is less than 2.5. To investigare this, she takes simple random sample of 150 households in the city and finds the sample mean number of people is 2.3 with standard deviation of 15, Can you conclude that the mean numnber of people per houschold is less than 2.5? Use confidence level of

Conduct a hypothesis test: For the past several years the mean number of people in household has been declining-A social scientist believes that in certain large ' city; the mean number of people per household is less than 2.5. To investigare this, she takes simple random sample of 150 households in the city and finds the sample mean number of people is 2.3 with standard deviation of 15, Can you conclude that the mean numnber of people per houschold is less than 2.5? Use confidence level of 9900 or _01 significance. Round your anstersto decimal places_ Wbat is the p-value?



Answers

Women Representatives in State Legislature A state representative wishes to estimate the mean number of women representatives per state legislature. A random sample of 17 states is selected, and the number of women representatives is shown. Based on the sample, what is the point estimate of the mean? Find the $90 \%$ confidence interval of the mean population. (Note: The population mean is actually $31.72, \text { or about } 32 .) \mathrm{Com}$pare this value to the point estimate and the confidence interval. There is something unusual about the data. Describe it and state how it would affect the confidence interval. $$ \begin{array}{ccccc}{5} & {33} & {35} & {37} & {24} \\ {31} & {16} & {45} & {19} & {13} \\ {18} & {29} & {15} & {39} & {18} \\ {58} & {132} & {}\end{array} $$

Now, this is the question that we have in this case. What is our end in his 20? So degree of freedom is going to become 19 inches 20 So D efs 19. This is the formula that I use for my standard deviation. Well, for by two. Since I'm constructing a 95% confidence interval, this is going to be 0.25 and I can again put in the values over here. I can find the value of the alphabet to using a table, substitute the values and I can get my confidence in trouble. Or I can simply use a software as I have done. Nobody care. Absolutely sorry. I have just simply put in the values. And this calculator has calculated all of the world is for me. You can see the sample mean the sample standard deviation on the confidence level of selected as 95. They can select whatever we want. A 2nd 95 and the conference of Devil s 4.96 to 8.44 point 964.96 to 8.4 This is my confidence interval now. Does it appear. The college students typically earned a bachelor degrees in four years. In four years. No, you can see that this confidence interval does not contain four years. So we can say that on an average means it is at 95% confidence. We can say that it takes more than 4.96 years to get a bachelor's degree. Is there anything about the data that would suggest that the confidence interval might not be a good result? Well, first of all, this is a very small sample size. This is a very small sample size, and the second thing is, uh, college Soon bachelor's degree based on data from Yeah, I think over here, the only problem is that the sample size is very small, so we cannot generalize it for the entire population.

Now let us look at this question. This is what we did in the last question. Let us read this one again. We're using the desert one more editor in appendix B, which includes 1 40 cent randomly selected adult females. And those ages have a standard deviation of 17.7. And here they're saying we able to assume Sigma also equal to 17.7. So in this case, this value is going to be 17.7 17.7. Okay. How many female statistics students ages must be obtained in order to estimate the mean age of all female statistics students. Okay, we want 95% confidence and that the sample mean it's within 1.5 years off. The population mean it's me. This is correct and we want 95% confidence this time. So this value of Z Alfa by two is going to change. What is going to be Alfa by two? It is going to be 0.0 to five and alphabet to Z Alpha by two is going to be 1.96 This value turns out to be 1.96 in this case 1.96 1.96 I just have to substitute the values I have Sigma I have zf of I do and I want to find out the value off n So the value of end is going to change here. And after doing this calculation, the value off and turns out to be 4815 and is equal to and is equal to four eat one fight. My N turns out to be 4815 And the next question is, does it seem reasonable to assume the ages of female statistics? Students have less variation? Yeah, you can say that the sample size is pretty large here, and we Can anybody see that by looking at this analysis?

We are very familiar with the methods. By now we want to construct the confidence level of 99 point Sorry off 99% and we have 20 different readings. So in order to control the confidence interval, what I've done is I've again simply input the values over here. These you can see we have 20 values. We are example mean a study 2.5 and a standard deviation is 20.3, you know, confidence level of 99. I select 99 this is the confidence interval that I get 20.8 to 44.2 20.8 20 pointed 20.8 to 44 point to This is my 99% confidence in trouble. Alright. Could also have done this by using the same old formula here and would have become 20 degree of freedom would be 19. Substitute these values Alfa. I would have become 0.50 point 00 fight and put the values in this formula and we come to the same result. This one now it is asked. Does this say? What is the question? There's the conference interval. Give us a good information about the population of all cancers off the same pretty brands that I consumed. No, because we have taken only one can off every brand in this, and only one can cannot be a representative off all the cans off that particular brand. So the answer is no. And we don't even care if this is normally distributed, because this is simply not going to give us a good idea off the, you know, off the mean off the mean off the mean of what we are. Yeah, the mean off caffeine for 12 rounds off a drink. So we don't even care if this is normally distributed.

Yeah. Alright, for this problem, we want to identify the null hypothesis, the alternative hypothesis testing six. This p value or critical values are daily. We just want to do something so we can come to a conclusion about our claim. So we're giving the following information that the sample size of 56 males are mean as 16,576 words. The size of 7870. The standard deviation is 7,871 words. And we're using a significant level of 0.01. We have a claim that mills have a higher, have a standard deviation is greater than the standard deviation of 7,476 words for females. So, first we wanted to find that no hypothesis so that we are h not our highlights, some key things and claims already. So we have the standard deviation. So that's what we're testing and we are testing that standard deviations Go to 707,460 words. So that's our no hypothesis. Our alternative. We're still testing the standard deviation, but that is greater than The 7460 words. So that's our no alternative hypothesis, nests are testes cystic. Since we're testing the standard deviation is going to be a chi squared and it's going to use the formula and minus one times S squared divided by stigma squared. Yeah. And luckily here, all this information is given to us. So we just plugged it in. We should get 61 point 227 next we can find the P value. Uh huh. Um You can find the P value either through the tables or with technology like our stack crunch. Um I personally I'm using are so if we're using our you can use a function p chi square. And what we're gonna play is gonna be the test at RN -1 or degrees of freedom. And that lower don't tell is equal to false. Yeah. So the reason why this whole as far lower dot tells equal to false is if normally we're doing a one tailed test, it will assume that it's true. That's our normal position for the peace chi squared. Well if we look back up here we're doing an upper tail test. So we want a slower detail to be called false. So if you plug this in you'll see that the p value of 0.2 64 Or 0.62 four. Sorry. Yeah. And just from the p value in apple, we can make conclusions. So remember if the p value is greater than alpha, we will acceptable have assists or if the p value it's greater than alpha, we will reject the null hypothesis. So if we look at our p value we have zero 2624 And we can see that this is greater than our alpha which is 0.01. Thus we can come to a conclusion that we will accept our no hypothesis or we will fail to reject are no hypothesis at Alpha equals 0.01. Yeah, so that the results is not statistically significant.


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