**Question 5**Early polls in the Georgia Senate race in 2022 showed an estimate of 49% votes for the Democrat candidate. Suppose that the estimated 95% confidence interval for the election results is [46%, 52%].a) Can you reject the null hypothesis that the true number of votes for the Democrat candidate is exactly 48% at a 5% significance level? And that the true value is exactly 44% at the same significance level?b) Do you have enough information to test whether the true number of votes for the Democrat candidate is exactly 47% at a 1% significance level? And that the true value is exactly 53% at that same significance level?c) Do you have enough information to test whether the true number of votes for the Democrat candidate is exactly 46.5% at a 10% significance level? And that the true value is exactly 45% at that same significance level?

It is given that: The estimate of the proportion of votes for a Democrat candidate, i.e., the sample proportion, p^=49=0.49. The 95% confidence interval for the true proportion of votes for the Democrat candidate is: 46%, 52%. Part (a) With 95% confidence, the proportion of votes for the Democrat candidate is estimated to be between 46% and 52% (46% and 52% are exclusive). It is hypothesized that: H0:p=48%=0.48 versus H1:p≠0.48 (two-tailed test). Considering the claimed proportion of votes of 48% (i.e., 0.48) lies within the estimated interval of [46%, 52%], the true proportion of votes at the population level is inferred to be indifferent to 48%. So the null hypothesis can not be rejected for the significance level of α=1-95%=0.05. Again it is hypothesized that: H0:p=44%=0.44 versus H1:p≠0.44 (two-tailed test). As the claimed proportion value of 44% does not lie within the estimated CI of [46%, 52%], the true proportion of votes is inferred to be not indifferent to the claimed proportion of 44%. With this inference, H0 is to be rejected. Part (b) For the 95% confidence level, the z-critical value is obtained using the excel function "NORM.S.INVprobability" as follows (where probability=1-α2; α=1-95%=0.05): So the z-critical value is 1.96. Using the sample proportion value of p^=0.49 and the CI of [0.46, 0.52], it is expressed that: Margin of Error=p^-Lower limit of CI =0.49-0.46⇒z×Standard Error (SE)=0.03⇒1.96×SE=0.03⇒SE=0.031.96 As the standard error and the estimated proportion of votes are known, there is sufficient information available to test the claim regarding the true value of the proportion of votes for any considered significance level. The z-critical value for α=1%=0.01 is obtained as follows: So the 99% CI (i.e., 1-α=1-1%=99%) for the true proportion of votes is estimated as: p^±z SE=0.49±2.5758×0.031.96≈0.49±0.0394≈0.49-0.0394, 0.49+0.0394≈0.4506, 0.5294 Considering the estimated interval includes the claimed proportion of 47% (i.e., 0.47) lies in the estimated interval of the proportion of votes between 0.4506 and 0.5294 (both the upper and lower limits are exclusive), with the same confidence level of 99% or significance level of 1%, the true proportion of votes at the population level may be concluded to be indifferent to 47%. As the claimed proportion of 53% (i.e., 0.53) does not lie in the estimated interval of the proportion of votes between 0.4506 and 0.5294 (both the upper and lower limits are exclusive), with the same confidence level of 99% or significance level of 1%, the true proportion of votes at the population level may not be concluded to be indifferent to 53%. Part (c) As the standard error and the estimated proportion of votes are known, there is sufficient information available to test the claim regarding the true value of the proportion of votes for any considered significance level. The z-critical value for α=10%=0.10 is obtained as follows: So the 90% CI (i.e., 1-α=1-10%=90%) for the true proportion of votes is estimated as: p^±z SE=0.49±1.6449×0.031.96≈0.49±0.02518≈0.49-0.02518, 0.49+0.02518≈0.46482, 0.51518 Considering the estimated interval includes the claimed proportion of 46.5% (i.e., 0.465) lies in the estimated interval of the proportion of votes between 0.46482 and 0.51518 (both the upper and lower limits are exclusive), with the same confidence level of 90% or significance level of 10%, the true proportion of votes at the population level may be concluded to be indifferent to 46.5%. As the claimed proportion of 45% (i.e., 0.45) does not lie in the estimated interval of the proportion of votes between 0.46482 and 0.51518 (both the upper and lower limits are exclusive), with the same confidence level of 90% or significance level of 10%, the true proportion of votes at the population level may not be concluded to be indifferent to 45%.