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Determine the sample size for the estimate of $mu$ for the following.a. $E=2.3, quad sigma=15.40$,confidence level $=99 %$b. $E=4.1, quad sigma=23.45$,confidence le...

Question

Determine the sample size for the estimate of $mu$ for the following.a. $E=2.3, quad sigma=15.40$,confidence level $=99 %$b. $E=4.1, quad sigma=23.45$,confidence level $=95 %$c. $E=25.9, quad sigma=122.25, quad$ confidence level $=90 %$

Determine the sample size for the estimate of $mu$ for the following. a. $E=2.3, quad sigma=15.40$, confidence level $=99 %$ b. $E=4.1, quad sigma=23.45$, confidence level $=95 %$ c. $E=25.9, quad sigma=122.25, quad$ confidence level $=90 %$



Answers

Construct the indicated confidence intervals for $(a)$ the population variance $\sigma^{2}$ and $(b)$ the population standard deviation $\sigma .$ Assume the sample is from a normally distributed population. $$c=0.95, s^{2}=11.56, n=30$$

The critical values from Table six using decrease of freedom equal to 17, which is in minus 1 80 minus one. So the chi square off one minus off over two is equal to 8.672 and the chi square off Alfa over two is equal to 27.5887 So the boundaries on the confidence and terrible for the standard deviation is and minus one over chi square off all for over 25 Standard deviation is 27.475 and in minus one over chi square off one minus off over two times. Ask is equal to 49.0, for the boundaries for the variance is this square value of this eh? Values put in 754 point 876 and 2401.392

Number 12. The degrees of freedom is April toe end minus one, which is 41 minus minus one. Eso the X squared of one minus uh, which is 22.164 and X squared off. Alfa is 63.691 There's for the standard deviation in square root off n minus one over X squared off. Over to which times? The standard deviation, which is equal to minus one or 63.691 times 278 0.1 which approximately equal toe 220.39 and and when this one over x squared one minus off over two times. Still a deviation, which is 41 minus 1/22 410.164 times 278 0.1, which approximately equal to 373.6. For the variance we get the the square off these boundaries, which is 224 3 19 square and 373.6 square, which is equal to 48 571.752 and 139576.96

Now, In this case, we want to generate a confidence interval where C is equal to 0.9 I will be having all of these values expires 12.3 expires 12.3, then signals 1.550 Sigma is 1.5 and is equal to 50. Now how do I find a confidence interval? The formula for confidence interval is expert plus minus the margin off error. What is the margin of error for the margin of error is E Alfa by two Sigma by route n this term this is known as margin off error. Now what is Alfa my C 0.9. So my Alfa becomes 0.1 and my Alfa by two is 0.5 Okay. And what I can do is I can use a Z value calculator, find my critical value of Z, and then substitute the values over here and get my confidence interval. Lower limit, comma, upper limit. Now, this is if I want to do it manually. But what if I want to do it using a calculator or using some using some technology? That is what I'm going to do over here. I have opened up this website. All right, now I want to take the help off this confidence interval calculator. What is my confidence? Level C is equal to 0.9. So this is going to be 0.9 or 90%. What does my expert, 12.3 my sample mean is 12.3, 12 point three. Okay. My sigma is 1.5 and s 50. Sigma is 1.5 1.5 and is 50 now. This is the confidence interval that I get. The confidence interval that I'm getting is 8.81 to 15.8, 8.81 to 15.8. This is my confidence interval. See, I am 90% confident that my two population being is going to lie between these values. This is my answer.

Now, the information that is given to me in this question is my C is 0.9. My C is 0.9, that is I want a 90% confidence interval which means my Alfa is going to be 0.1 or Alfa by two will become 0.5 My sample minutes 12.3 My sigma, that is my population. Standard deviation is 1.5 and my end or sample size is 50. Now, in order to calculate the confidence interval, this is the formula that we use Where said Alfa by two is the critical value of Z for this particular value off Alfa. By 20.5 we can apply this formula and get our confidence interval lower limit comma, upper limit. Or I can use a calculator that is technology in order to find this which is what we will be doing in most of the real world examples anyway. So let's just open up a Web site that is going to help us to calculate the confidence interval over here. You see, I'm using this website and I have just put in the values like my sample mean my sample size, my standard deviation and my confidence level. So the result that I'm getting is 11.95 to 12.64 This is my confidence Interval 11.95 11.95 to 12.64 This is my confidence interval. Which means I am 90% confident that my true population mean is going to lie in this interval. This is my answer.


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