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Problem [26 points total][9 points] Suppose we have a random sample Xi, _, Xn from N(u,02) distribution; where 02 is a known constant. Consider the Z-test of Ho: p ...

Question

Problem [26 points total][9 points] Suppose we have a random sample Xi, _, Xn from N(u,02) distribution; where 02 is a known constant. Consider the Z-test of Ho: p = Ho versus the following hypotheses with corresponding critical regions:Hypothesis Ha: u > Lo Ha: H < |o Ha: H # LoCritical region C = {(xX1, Xn): 2 > 2(1 - a)} {(x1, ,Xn): 2 < -2(1 - 0)} {(x1_,xn):Izl > 2(1 - a/2 )}1) ii) iii)where the critical regions are defined in terms of the test statistic Z = (X _ /o)/- J (see s

Problem [26 points total] [9 points] Suppose we have a random sample Xi, _, Xn from N(u,02) distribution; where 02 is a known constant. Consider the Z-test of Ho: p = Ho versus the following hypotheses with corresponding critical regions: Hypothesis Ha: u > Lo Ha: H < |o Ha: H # Lo Critical region C = {(xX1, Xn): 2 > 2(1 - a)} {(x1, ,Xn): 2 < -2(1 - 0)} {(x1_,xn):Izl > 2(1 - a/2 )} 1) ii) iii) where the critical regions are defined in terms of the test statistic Z = (X _ /o)/- J (see slides 1.5). For each case, carefully show that the test has significance level & using probability calculations_ b) [5 points] Using the critical region from the test of the hypotheses in part (a i) , derive a one-sided 90% confidence interval for U. [5 points] Using the critical region from the test of the hypotheses in part (a ii), derive a one ~sided 90% confidence interval for U. [5 points] Using the critical region from the test of the hypotheses in part (a iii) , derive the two-sided 90% confidence interval for U. Is it wider O narrower than the confidence interval you found in problem (6)2 [2 points] Provide the interpretation of a (1 0)100% confidence interval for p_



Answers

The confidence interval for the median of a set of values less than or equal to 25 in number can be found by ordering the data from smallest to largest, finding the median, and using Table J. For example, to find the $95 \%$ confidence interval of the true median for $17,19,3,8,10,15,1,23,2,12,$ order the data: $$ 1,2,3,8,10,12,15,17,19,23 $$ From Table $\mathrm{J}$, select $n=10$ and $\alpha=0.05,$ and find the critical value. Use the two-tailed row. In this case, the critical value is $1 .$ Add 1 to this value to get $2 .$ In the ordered list, count from the left two numbers and from the right two numbers, and use these numbers to get the confidence interval, as shown: $$ \begin{array}{l}{1,2,3,8,10,12,15,17,19,23} \\ {2 \leq \mathrm{MD} \leq 19}\end{array} $$ Always add 1 to the number obtained from the table before counting. For example, if the critical value is $3,$ then count 4 values from the left and right. For Exercises 21 through 25 , find the confidence interval of the median, indicated in parentheses, for each set of data. $$ 3,12,15,18,16,15,22,30,25,4,6,9(95 \%) $$

Problem. 24. The order. The native values from the smallest to the launches, which is one, 235 and six eight and 10, 15 and 15, 21 24 31 33 41 42 54 and 56 58 and 65 determined the critical value in Table Jane. He was in Alpha's April 2 opening toe all one and then is equal to 19, which is stale, uh, the two tails. So the critical value is April 23 and case equal to the critical value across one which is equal force through the boundaries for the confidence interval for the meeting are then the forces smallest value and the force largest value. So the median funds between five and 54.

Brooklyn 25 order the data values from the smallest to the largest, just 12 and 14. And 14, 15 and 16 on 17 and 18. And 19, 19 and 21 23 25 27 32 33 35 39 41 42 47. So determined The critical values Imperial and J, using up as equal to 0.5 and then is able to train. So the critical values is even toe find. So K is equal to the critical values plus one. So five plus one, she's six and the boundaries off the confidence in terms for the meeting or then the fix. Smallest value and the sixth largest value. Eso 123456 So the median inbounded between 17 and 33.

Problem 22 or with the data values from the smallest to the largest, which is hundreds, hundreds on one 1/6. Then 115 115 141 one or two and one for three and 14 green and one for five and one for seven and one for seven and 1 15 and 1 52 on 1. 53 and 1 55 and 1 57 and 1 61 63 and 1 64. So to remind the critical value using Table and J, who's offer is equal to opening one and then is equal to 20 so the critical value is equal to find, so add one toe critical. Various okay is equal to five plus one, which is six to the boundaries, off the confidence interval for the million or then six smallest and the sixth largest value insulting data list, which is 1 41 and 1 53

Problem. 23. The order off the data values from the smallest to the largest is 4.24 point five on 4.74 point eight and 5.1 and 5.2 and 5.6 and 6.3, 6.3 and 7.1, 7.2, 7.8 and 8.2, 9.3 and 9.3 and 9.5 on 9.6. So your mind, uh, the critical value in In table using Table J and Alfa is equal to offering to go to and then is equal to 16 toe failed eso. The critical value is equal to two, so the K is equal toe the critical value plus one which is equal to three. So the boundaries off the confidence interval for the median are in the third smallest and the third largest very in the Sorted data, which is a 4.7 and 9.3


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