91Ó°ÊÓ

Involve finding binomial probabilities, finding parameters, and determining whether values are significantly high or low by using the range rule of thumb and probabilities. In a presidential election, 611 randomly selected voters were surveyed, and 308 of them said that they voted for the winning candidate (based on data from ICR Survey Research Group). The actual percentage of votes for the winning candidate was \(43 \%\). Assume that \(43 \%\) of voters actually did vote for the winning candidate, and assume that 611 voters are randomly selected. a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the 308 voters who said that they voted for the winner significantly high? b. Find the probability of exactly 308 voters who actually voted for the winner. c. Find the probability of 308 or more voters who actually voted for the winner. d. Which probability is relevant for determining whether the value of 308 voters is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 308 voters who said that they voted for the winner significantly high? e. What is an important observation about the survey results?

Step by step solution

01

Understand the problem statements and given parameters

We are given that 611 voters were surveyed, and 308 said they voted for the winning candidate. The given voting percentage for the winning candidate is 43%.

02

Calculate mean and standard deviation

The mean \(\mu\) and the standard deviation \(\sigma\) for a binomial distribution are calculated as \[ \mu = np \] \[ \sigma = \sqrt{np(1-p)} \] where n = 611 and p = 0.43. \(\) = 611 \(p\) = 0.43. \[\mu = 611 \times 0.43 = 262.73\] \[\sigma = \sqrt{611 \times 0.43 \times 0.57} = 12.3 \]

03

Use the range rule of thumb to identify limits

The range rule of thumb states that values are significantly low if they are below \(\mu - 2\sigma\) and significantly high if they are above \(\mu + 2\sigma\). Calculate these limits: \[\mu - 2\sigma = 262.73 - 2 \times 12.3 = 238.13 \] \[\mu + 2\sigma = 262.73 + 2 \times 12.3 = 287.33 \]

04

Check if 308 is significantly high or low

Since 308 is above the upper limit of 287.33, it is considered significantly high.

05

Calculate the probability of getting exactly 308 voters

Using the binomial probability formula \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n = 611, k = 308, p = 0.43 \), we have: \[ P(X = 308) = \binom{611}{308} 0.43^{308} (1-0.43)^{611-308} \] This calculates to be approximately 0.00089.

06

Calculate the probability of getting 308 or more voters

To find \[ P(X \geq 308) \] sum all probabilities from 308 to 611. \[ P(X \geq 308) = 1 - P(X < 308) \] Using cumulative binomial probability tables or software, this calculates to be approximately 0.00098.

07

Identify the relevant probability for significance

The probability from part (c) is more relevant. If the probability is very low (usually less than 0.05), it suggests statistical significance.

08

Determine if 308 voters is significantly high

Since the probability 0.00098 is much lower than 0.05, the result of 308 voters is significantly high.

09

Consider an important observation about the survey

An important observation is that significantly more people reported voting for the winning candidate than expected. This might indicate response bias or inaccuracies in the survey.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range Rule of Thumb

The range rule of thumb is a straightforward guideline used in statistics to determine whether a value is unusually high or low compared to the mean (average). This rule states that values are significantly low if they are below \( \mu - 2\sigma \) and significantly high if they are above \( \mu + 2\sigma \). Here, \( \mu \) is the mean and \( \sigma \) is the standard deviation.

For example, in our exercise, the mean (\( \mu \)) is calculated as \(262.73\) and the standard deviation (\( \sigma \)) as \(12.3\). Using these values:

\( \mu - 2\sigma = 262.73 - 2 \times 12.3 = 238.13 \)
\( \mu + 2\sigma = 262.73 + 2 \times 12.3 = 287.33 \)

So, any value below 238.13 is significantly low and any value above 287.33 is significantly high. Since 308 is above 287.33, it is considered significantly high.

Statistical Significance

Statistical significance helps us decide if an observed effect or result is likely to be due to chance or if it genuinely reflects a real phenomenon. In practical terms, we often use a threshold known as a p-value to determine this.

If the p-value is less than our significance level (commonly 0.05), we conclude that our result is statistically significant.

In our exercise, we computed the probability of exactly 308 voters voting for the winner as \(P(X = 308) = 0.00089\) and the probability of 308 or more voters as \(P(X \geq 308) = 0.00098\). Since 0.00098 is much lower than 0.05, we see that 308 voters is significantly high.

Statistical significance is critical in hypothesis testing and helps us infer whether an observed outcome is unusual or aligns with what we would expect by random variation alone.

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials of a binary experiment. Each trial has two possible outcomes: success (\( p \)) and failure (\( 1 - p \)).

The mean (\( \mu \)) and standard deviation (\( \sigma \)) for a binomial distribution are given by:

\( \mu = np \)
\( \sigma = \sqrt{np(1-p)} \)

In our example, we have 611 voters surveyed (\( n = 611 \)) and a success probability of 43% (\