91影视

It is given that 43% of the voters did not vote for the winning candidate.

A sample of 611 voters is randomly selected. Out of 611, 308 voters said that they voted for the winning candidate.

The total number of voters selected (n) is equal to 611.

The probability of selecting a voter who actually voted for the winning candidate is as follows:

$$\begin{gathered} p=43\% \\ =\frac{43}{100} \\ =0.43 \end{gathered}$$

The mean of the number of voters who voted for the winning candidate is given as follows:

$$\begin{gathered} 渭=np \\ =611\left(0.43\right) \\ =262.73 \end{gathered}$$

Thus, $渭=262.73$.

The standard deviation for the number of voters who voted for the winning candidate is computed below:

$$\begin{gathered} 蟽=\sqrt{npq} \\ =\sqrt{np\left(1-p\right)} \\ =\sqrt{611脳0.43脳\left(1-0.43\right)} \\ =12.24 \end{gathered}$$

Thus, $蟽=12.24$.

a.

The following limit separates significantly low values:

$$\begin{gathered} 渭-2蟽=262.73-\left(2\right)\left(12.24\right) \\ =238.25 \\ 鈮�238.3 \end{gathered}$$

Therefore, values less than or equal to 238.3 are considered to be significantly low.

The following limit separates significantly high values:

$$\begin{gathered} 渭+2蟽=262.73+\left(2\right)\left(12.24\right) \\ =287.21 \\ 鈮�287.2 \end{gathered}$$

Therefore, values greater than or equal to 287.2 are considered to be significantly high.

Values lyingbetween238.3 and 287.2 are not significant.

Here, the value of 308 lies beyond 287.2Thus, it is considered significantly high.

b.

Let X denote the number of voters who actually voted for the winning candidate.

Success is defined as selecting a voter who voted for the winning candidate.

The probability of success is equal to p=0.43.

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.43 \\ =0.57 \end{gathered}$$

The number of trials (n) is equal to 611.

The binomial probability formula used to compute the given probability is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

By using the binomial probability formula, the probability of getting 308 voters who actually voted for the winning candidate can be calculated in the following manner:

$$\begin{gathered} P\left(X=308\right)={}^{611}{C}_{308}{\left(0.43\right)}^{308}{\left(0.57\right)}^{611-308} \\ =0.0000369 \end{gathered}$$

Thus, the probability of getting 308 voters who actually voted for the winning candidate is equal to 0.0000369.

c.

The probability of 308 or more voters who actually voted for the winning candidate has the following expression:

$P\left(X猢�308\right)=P\left(X=308\right)+P\left(X=309\right)+......+P\left(X=611\right)$

The individual probabilities will be computed as follows:

$$\begin{gathered} P\left(X=308\right)={}^{611}{C}_{308}{\left(0.43\right)}^{308}{\left(0.57\right)}^{303} \\ 鈮�0.0000369 \\ P\left(X=309\right)={}^{611}{C}_{309}{\left(0.43\right)}^{309}{\left(0.57\right)}^{302} \\ 鈮�0.0000273 \\ . \\ . \\ . \\ P\left(X=611\right)={}^{611}{C}_{611}{\left(0.43\right)}^{611}{\left(0.57\right)}^0 \\ 鈮�1.12脳{10}^{-224} \end{gathered}$$

Thus, the required probability is computed as follows:

$$\begin{gathered} P\left(X猢�308\right)=P\left(X=308\right)+P\left(X=309\right)+.......+P\left(X=611\right) \\ =0.0001357 \end{gathered}$$

Thus, the probability of 308 or more voters who actually voted for the winning candidate is equal to 0.0001357.

d.

The following is the probability formula for determining if a given number of successes (x) is significantly high or not:

$P\left(x\text{or}\text{more}\right)猢�0.05$

Here, the considered value of x is equal to 308.

Since the probability of 308 or more voters who actually voted for the winning candidate is represented in part (c), theprobability computed in part (c) is relevant for determining whether the result of 308 voters is significantly high.

Thus,

$$\begin{gathered} P\left(308\text{or}\text{more}\right)=0.0001357 \\ <0.05 \end{gathered}$$

Since the probability of 308 or more voters who actually voted for the winning candidate is less than 0.05, the value of 308 voters who voted for the winning candidate is significantly high.

e.

Since, out of 611 voters, the probability of 308 voters who actually voted for the winning candidate is very small, it indicates that a substantial number of voters are probably lying that they had voted for the winning candidate.