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According to the given information, it can not be claimed that at least half of all voters prefer democrat because of lack of sufficient evidence as observed from the p-value for the corresponding test statistic or z-score.
It is given to us that -
A poll of 1068 adult Americans is carried out
The poll reveals that 48% of the votes surveyed prefer the democratic candidate for the presidency
The significance level of the poll survey is 0.05
We have to test the claim that at least half of all voters prefer democrat.
From the given information, we have
Sample size, n = 1068
Sample proportion, p = 48% = 0.48
Significance level, α = 0.05
Hypothesized proportion value, P = 50% = 0.5
Firstly, let us define the null hypothesis and the alternative hypothesis. According to the information, we can say that -
Null hypothesis, [tex]H_{0} = 0.05[/tex]
Alternative hypothesis, [tex]H_{A} < 0.05[/tex]
We know that the test statistic is referred to as the z-score and the formula for this test statistic is given as -
[tex]z = \frac{p-P}{\sqrt{\frac{P(1-P)}{n} } }[/tex] ------ (1)
Substituting the values of p, P, and n from the given information in equation (1), we have
[tex]z = \frac{p-P}{\sqrt{\frac{P(1-P)}{n} } }\\= > z = \frac{0.48-0.5}{\sqrt{\frac{0.5(1-0.5)}{1068} } }\\= > z = \frac{-0.02}{0.01529} \\= > z = -1.31[/tex]
For z = -1.31, the corresponding p-value as calculated from the z-score table is 0.951, which is more than hypothesized value of proportion.
Thus, we can conclude that it can not be claimed that at least half of all voters prefer democrat because of lack of sufficient evidence as observed from the p-value for the corresponding test statistic or z-score.
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