Question

Suppose that \(90 \%\) of all registered California voters favor banning the release of information from exit polls in presidential elections until after the polls in California close. A random sample of 25 California voters is to be selected. a. What is the probability that more than 20 voters favor the ban? b. What is the probability that at least 20 voters favor the ban? c. What are the mean value and standard deviation of the number of voters who favor the ban? d. If fewer than 20 voters in the sample favor the ban, is this at odds with the assertion that (at least) \(90 \%\) of the populace favors the ban? (Hint: Consider \(P(x<20)\) when \(\pi=.9 .)\)

Short answer

The solutions are as follows: \n a. The probability that more than 20 voters favor the ban = the sum of probabilities for x=21, 22, 23, 24, and 25. \n b. The probability that at least 20 voters favor the ban = the sum of all probabilities for x=20, 21, 22, 23, 24, and 25. \n c. Mean is np = 22.5 voters, and σ is \( \sqrt{22.5 \cdot 0.1}=1.5 \) voters. \n d. Calculate \( P(x < 20) \). It is used to measure the statistical odd of our assertion that 90% support the ban. It's a measure used for contradicting or supporting our assertion.

Step-by-step solution

Identify Variables

In this problem, we have the following variables: number of trials (n), number of successes (x), and the probability of success (p). n=25, x varies based on the question part, and p=0.9 (since 90% of the population supports the ban).

Probability of More Than 20 Voters Favoring the Ban

We need to calculate \( P(x > 20) \). Since it's more than 20, it includes 21, 22, 23, 24 and 25. Each of these is calculated with the binomial probability formula: \( P(x) = C(n,x) \cdot (p^x) \cdot (1-p)^(n-x) \). Then sum these probabilities.

Probability of At Least 20 Voters Favoring the Ban

Now, we need to calculate \( P(x \geq 20) \). It includes 20, so we calculate for x=20, 21, 22, 23, 24, and 25 in the same way as in Step 2. Then sum these probabilities.

Calculate the Mean and Standard Deviation

The mean (µ) and standard deviation (σ) for a binomial distribution are calculated as µ=np and σ= \( \sqrt{np(1-p)} \). Plug in the given values and calculate each.

Interpret the Probability for x

We calculate the probability \( P(x < 20) \) using the same formula as in step 2 and 3. This value would statistically indicate whether our assertion of 90% support is consistent or not. If the probability is too high, it may suggest that our assertion may not be accurate and vice versa.

Key concepts

Probability of Success

Understanding the 'probability of success' is crucial when dealing with binomial distribution. In the context of our exercise, the 'success' in question refers to a California voter favoring the ban on releasing exit poll information. The given probability of this success is 90%, which is symbolically represented as a probability (p) of 0.9. This figure is incredibly important because it will be the basis for all subsequent probability calculations. The probability of success for each trial, or voter in this case, is assumed to be independent, meaning one voter's stance does not affect another's, and consistent across all trials in the sample.

Students often confuse 'probability of success' with the actual occurrence of successes. It's vital to remember that while the probability might be high, it doesn't guarantee a high number of successes. For example, even with a 90% probability, it's possible (though unlikely) that a sample may yield fewer supporters than expected, which is why understanding binomial probability distribution is essential.

Binomial Probability Formula

Calculating Individual Probabilities

With the probability of success in hand, we move to using the binomial probability formula to calculate the likelihood of different outcomes. The binomial probability formula is given by: \[ P(x) = C(n,x) \cdot (p^x) \cdot (1-p)^(n-x) \] where \( P(x) \) is the probability of exactly \( x \) successes in \( n \) trials, \( C(n,x) \) is the combination of \( n \) items taken \( x \) at a time, \( p \) is the probability of success, and \( 1-p \) is the probability of failure for each individual trial. For our exercise, this formula helps to calculate the exact probability of getting more than 20 out of 25 voters in favor of the ban, as well as for getting at least 20 voters.

Summing Up Probabilities

When calculating for 'more than 20' or 'at least 20,' you'll often need to calculate individual probabilities for 21, 22, 23, 24, and 25 (or include 20, respectively), each using the formula, and then sum them up. This is because these probabilities are mutually exclusive events; they cannot happen at the same time in one trial but are part of the total probability for 'more than 20' or 'at least 20' voters. Understanding and accurately applying this formula is pivotal in solving binomial distribution problems.

Mean and Standard Deviation of Binomial Distribution

When grappling with binomial distribution, two key statistical measures to understand are the mean and the standard deviation. The mean of a binomial distribution, denoted as \( \mu \), gives the expected number of successes in a given number of trials and is calculated as \( \mu = np \). In our case, with a sample of 25 voters and the success probability of 0.9, the mean tells us the expected number of voters in favor of the ban.

To measure the variability or dispersion from this mean, we calculate the standard deviation using the formula \( \sigma = \sqrt{np(1-p)} \). This standard deviation provides insight into the expected range of the number of successes from the mean value. In the exercise's context, by calculating the standard deviation, we can understand the spread or the consistency of the voters' support regarding the ban. Knowing both the mean and standard deviation is instrumental for interpreting the results of the binomial experiment and for assessing any assertions made about the population as a whole, like the claim that 90% of the populace favors the ban.