Question

Table of Responses - X Yes - have reduced Yes - not been able to No - not certain Total Year 1 300 300 300 900 Year 2 293 200 407 900 Total 593 500 707 1,800 Print Done OTTE e not independent not equal

Answer 1

(A) Null and the alternate hypothesis

Option "D" is correct.

H0 : The response is independent of year

H1 : The response in not independent of year.

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The test statistic is 36.276

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The p-value is 0

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Conclusion : Null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are not independent.

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The following cross tablulation have been provided. The row and column total have been calculated and they are shown below: Column 1 Column 2 Total 300 293 593 300 200 500 300 407 707 Total 900 900 1800 The expected values are computed in terms of row and column totals. In fact, the formula is Ejj = Rixos, where R; corresponds to the total sum of elements in row i, C; corresponds to the total sum of elements in column j, and T is the grand total. The table below shows the calculations to obtain the table with expected values: Expected Values Column 1 Column 2 Total 593 900x593 296.5 1800 900X500 = 250 1800 900x707 = 353.5 1800 900 X 593 296.5 1800 900x500 1800 = 250 900X 707 = 353.5 1800 500 707 Total 900 900 1800

Based on the observed and expected values, the squared distances can be computed according to the following formula: (E-0)?/E. The table with squared distances is shown below: Squared Distances Column 1 Column 2 0.041 0.041 (300-296.5) 296.5 (300250 250 (300-353.5) 353.5 (293–296.5) 296.5 (200-250 250 = 10 = 10 = 8.097 (407-353.5) 353.5 8.097 Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: The two variables are independent H: The two variables are dependent This corresponds to a Chi-Square test of independence. Rejection Region Based on the information provided, the significance level is a = 0.05, the number of degrees of freedom is df = (3-1) X (2-1) = 2, so then the rejection region for this test is R= +{x? :x> 5.991}

Test Statistics The Chi-Squared statistic is computed as follows: n = (Oi; – E;;)? Ejj = 0.041 +10+ 8.097 + 0.041 +10+ 8.097 = 36.276 i=1 Decision about the null hypothesis Since it is observed that x² = 36.276 > xã = 5.991, it is then concluded that the null hypothesis is rejected. Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level. The corresponding p-value for the test is p = Pr(xź > 36.276) = 0.

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(b) When population proprtion for year 1 is p1 and population proportion for year 2 is p2

The Null and the alternate hypothesis is given as

Option "F" is correct

H0 : p1-p2 = 0

Ha : p1-p2  ≠ 0

Since Level of significance is not visible in the picture,

So, we assume it to be 0.05

To reject H0,

Select option "B"

Reject H0 : for z < -1.96 or z > 1.96

The z score is 0.351

Do not reject H0, There are not enough evidence to claim that there is significant difference between the population proportion for year 1 and year 2

300 900 For sample 1, we have that the sample size is Ni = 900, the number of favorable cases is X1 = 300, so then the sample proportion is pi = x = 0.3333 For sample 2, we have that the sample size is N2 900, the number of favorable cases is X2 = 293, so then the sample proportion is P2 X = 293 = 0.3256 The value of the pooled proportion is computed as = x +X 900-900 = = 300—293 = 0.3294 = Also, the given significance level is a = 0.05. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Pi = P2 Ha: p = P2 This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a two- tailed test is ze = 1.96. The rejection region for this two-tailed test is R= {z : 21 > 1.96}

(3) Test Statistics The z-statistic is computed as follows: Z= P1 - 09 Vp(1-0)(1/T1 +1/09) = 0.3333 – 0.3256 0.3294 . (1 - 0.3294)(1/900+ 1/900) = 0.351 (4) Decision about the null hypothesis = 1.96, it is then concluded that the null hypothesis is not Since it is observed that z1 = 0.351 <Zc rejected. Using the P-value approach: The p-value is p= 0.7256, and since p= 0.7256 > 0.05, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pı is different than p2, at the 0.05 significance level. Confidence Interval The 95% confidence interval for pi - p2 is: -0.036 <p1 - p2 < 0.051.