A political pollster is conducting an analysis of sample results in order to make predictions on election night.Assuming a two-candidate election,if a specific candidate receives at least 54%of the vote in the sample,that candidate will be forecast as the winner of the election.You select a random sample of 100 voters. Suppose that the sample size was increased to 400.Repeat process(a)through(c),using this new sample size.Comment on the difference. The probability isthat a candidate will be forecast as the winner when the popuio percentage of her vote is50.1%. (Round to four decimal places as needed.)
Question
A political pollster is conducting an analysis of sample results in order to make predictions on election night.Assuming a two-candidate election,if a specific candidate receives at least 54%of the vote in the sample,that candidate will be forecast as the winner of the election.You select a random sample of 100 voters. Suppose that the sample size was increased to 400.Repeat process(a)through(c),using this new sample size.Comment on the difference. The probability isthat a candidate will be forecast as the winner when the popuio percentage of her vote is50.1%. (Round to four decimal places as needed.)
Solution
To solve this problem, we can use the Central Limit Theorem which states that if we have a sufficiently large sample size, the sampling distribution of the mean will be approximately normally distributed.
The mean (μ) of the sampling distribution of the mean is equal to the population proportion, which is 50.1% or 0.501.
The standard deviation (σ) of the sampling distribution of the mean, also known as the standard error, is calculated using the formula sqrt[p(1-p)/n], where p is the population proportion and n is the sample size. So, σ = sqrt[0.501*(1-0.501)/400] = 0.025.
We want to find the probability that the sample proportion is at least 54% or 0.54. To do this, we can standardize 0.54 using the Z-score formula:
Z = (X - μ) / σ
Where X is the value we are interested in, μ is the mean, and σ is the standard deviation. Plugging in the values we have:
Z = (0.54 - 0.501) / 0.025 = 1.56
Now, we can look up this Z-score in the Z-table to find the probability. The value for Z = 1.56 is 0.9406. However, since we are looking for the probability of getting at least 54%, we need to subtract this value from 1.
So, the probability that a candidate will be forecast as the winner when the population percentage of her vote is 50.1% is 1 - 0.9406 = 0.0594 or 5.94%.
Increasing the sample size from 100 to 400 decreased the standard error, which made the sampling distribution more concentrated around the mean. This means that the probability of getting a sample proportion of at least 54% decreased when the sample size increased.
Similar Questions
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