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A 95% confidence interval forpis found to be [.41,.62]. Test thefollowing hypotheses,or explain why you can’t.a.H0: p=.5 at significance levelα=.05.b.H0: ...

Question

A 95% confidence interval forpis found to be [.41,.62]. Test thefollowing hypotheses,or explain why you can’t.a.H0: p=.5 at significance levelα=.05.b.H0: p=.65 at significance level α=.05.c.H0: p=.65 at significance level α=.01.d.H0: p=.65 at significance levelα=.10.

A 95% confidence interval forpis found to be [.41,.62]. Test the following hypotheses,or explain why you can’t. a.H0: p=.5 at significance levelα=.05 .b.H0: p=.65 at significance level α=.05. c.H0: p=.65 at significance level α=.01. d.H0: p=.65 at significance levelα=.10.



Answers

Which of the following 95$\%$ confidence intervals would lead us to reiect $H_{0} : p=0.30$ in favor of
$H_{a} : p \neq 0.30$ at the 5$\%$ significance level?
$$
\begin{array}{l}{\text { (a) }(0.29,0.38) \quad \text { (c) }(0.27,0.31) \quad \text { (e) None of these }} \\ {\text { (b) }(0.19,0.27) \quad \text { (d) }(0.24,0.30)}\end{array}
$$

Okay. Okay. So we're doing confidence intervals. It's this formula right here we're doing it at the 5% and the 1% significance level. So for tea at the 5% significance level we have that it is a critical value of 2.145 Mhm. This is going to turn out to get us. And points of 0.28 and 41 832 at the alpha equals 5% significance level. Now of course we can repeat this for the other significance level of point A. One. And we see that the critical value adds T point oh five is 2.977 Yeah. So there, yeah. Mhm. So that gets us a lower bound of negative 8079 And our upper bound is going to be 49 939 This is our 99% confidence interval and there we go.

Okay, so we're doing a 90% confidence interval which means a significant level of 10% and the other ones at 98%. So we have a 2% significance level for the second one. Our values is calculated from problem for 83. Page 483 on 10.142 are going to be as follows. In red here, Our degrees of freedom is one less than the sample size, which is nine. And in table four we will be able to find out what the critical value is. So in nine degrees of freedom. And at the 10% significance level we find that it is 1.833 Which gives us an upper bound, sorry, a lower bound of 1.17 And a lower bound of 3.49. This is going to be at the 10% significance level. So this right here is our 90% confidence interval Now, as for our 98% confidence interval. The only thing that's going to change this t critical value, which is 2.8-1, And then that gives us a lower bound of 0.544 for And an upper bound of 4.116. And again, this is that the 2% significance level, so this is our 98% confidence interval.

So for this problem here we're going to find the confidence level. Okay. Yeah. And the significance level For two different confidence intervals. Yeah. Okay, so for a we have a confidence confident interval. Mhm Yes. of 85%. So to find its confidence level we're just moving the decimal over two places And that's going to make it be 0.85. Yeah. Yeah. And to find its significance level we also denoted as alpha. I'm not really good at drawing those but something like this. And so alpha is going to be actually 1 -1 my confidence level. So that's going to be 0.15. Yeah. And that's going to be my significant level. And then for B it gives us a confidence interval of 95%. So again to find the confidence level we are moving the decimal over two places so that 0.95. And then to find its significance level I do 1 -1 my confidence level. So 1 -0.95. Yeah. Yeah. Which gives me 0.05. And that is my significance level. Yeah. Yeah.

Which level of confidence is going to produce the brightest confidence interval. There are four options. The first one is 90. Then we have 95 98 99. Now what exactly is a confidence interval? Let's say this is my normal distribution. Let's say I'm marking these two critical values for 90 percent. Okay, let's say that these are Z one plus minus Z one. These are for 90%. 90% means the remaining 10%. The area off 0.1 is going to lie in these peel's. Okay, so now by this logic, if I say 95% would these bars shift? Outwards are in words. Let's just think we want 95% confidence interval, which means we are only 5% or this area. And the tales, Toby 0.5 which means we want this area to reduce. So when I say 95% what is going to happen is these bars are going to shift outwards. These bars will shift outwards. So as my confidence level increases, these bars keep on shifting outwards, which means that my interval gets wider and wider. So what option is the biggest 99% 99% means that there is only zero point 01 area in these tales. In both of these tales, my area as a 0.1 which means this middle area is going to be 0.99 right, 0.99 So this is going to be my Vitus interval. So my answer this question is going to be option number T 99%.


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