Part 1:

You are given that 31% of likely voters are willing to vote by internet (i.e., $p = 0.31$), and a sample of 12 likely voters is randomly selected. This situation can be modeled using the binomial distribution, where the probability of getting exactly $k$ successes (internet voters) in $n$ trials (voters) is given by the binomial probability formula:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Where:

• $n = 12$ (number of trials),
• $k = 9$ (number of successes),
• $p = 0.31$ (probability of success).

We need to calculate $P(X = 9)$.

$P(X = 9) = \binom{12}{9} (0.31)^9 (0.69)^{3}$

First, calculate the binomial coefficient:

$\binom{12}{9} = \frac{12!}{9!(12-9)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220$

Now, calculate the probability:

$P(X = 9) = 220 \times (0.31)^9 \times (0.69)^3$

Using a calculator:

$P(X = 9) \approx 220 \times 0.000023373 \times 0.328509$

$P(X = 9) \approx 0.00276$

So the probability that exactly 9 of the selected voters would do internet voting is approximately 0.00276.

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Part 2:

To determine if the value of 9 is "significantly high," we need to assess the probability of getting 9 or more voters who would vote by internet. This means we need to calculate the cumulative probability $P(X \geq 9)$, which is the sum of $P(X = 9)$, $P(X = 10)$, $P(X = 11)$, and $P(X = 12)$.

However, based on the small probability of $P(X = 9) = 0.00276$, it suggests that getting exactly 9 is quite rare. We need to calculate the cumulative probability to determine if it is "low."

First, calculate the probabilities for $P(X = 10)$, $P(X = 11)$, and $P(X = 12)$, and sum them.

If the cumulative probability is low (typically below a threshold of 0.05), it would suggest that 9 is significantly high. Otherwise, it is not.

Given the very small value for $P(X = 9)$, it's likely that the cumulative probability for $X \geq 9$ will be low.

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Answer Choice:

D. No, because the probability of 9 or more is low. (The probability is small, indicating that 9 is quite high relative to what we would expect in this binomial distribution.)

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Additional Notes:

If you'd like to confirm the exact cumulative probability or need further assistance with the calculations, let me know!

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Follow-up Questions:

1. How do you calculate binomial probabilities for a different sample size or probability?
2. What is the mean and standard deviation of a binomial distribution?
3. How would the probability change if the percentage of internet voters was 50%?
4. What does the "low probability" threshold of 0.05 mean in hypothesis testing?
5. How do you interpret a binomial probability in real-world contexts like elections?

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Tip: When solving binomial probability problems, make sure to check whether the conditions (fixed number of trials, two possible outcomes, constant probability) are satisfied to use the binomial distribution.