High School

In the upcoming election for governor, the most recent poll, based on 900 respondents, predicts that the incumbent will be reelected with 55 percent of the votes.

For the sake of argument, assume that 51 percent of the actual voters in the state support the incumbent governor \((p = .51)\).

Calculate the probability of observing a sample proportion of voters .55 or higher supporting the incumbent governor.

Answer :

The probability is calculated to be 0.8%.

To solve this, we need to use the concept of the sampling distribution of the sample proportion.

We will use the following steps:

  1. Identify the population proportion (p = 0.51) and sample size (n = 900).
  2. Calculate the standard error (SE) of the sample proportion using the formula:

SE = √[p(1-p)/n]

Substituting the values:

SE = √[0.51 * 0.49 / 900] = √[0.2499 / 900] ≈ 0.0166

  1. Find the z-score for the sample proportion (0.55). The z-score formula is:

z = (p' - p) / SE

Where p' is the sample proportion:

z = (0.55 - 0.51) / 0.0166 ≈ 2.41

  1. Use the z-score to find the probability from the standard normal distribution table.
  2. A z-score of 2.41 corresponds to a probability of 0.9920.

Therefore, the probability of observing a sample proportion of 0.55 or higher is:

1 - 0.9920 = 0.0080

Thus, the probability is 0.8%.

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