A student government representative at a local university claims that 60% of all the undergraduate students favor a move to Division I in college football. A simple random sample of 250 undergraduate students is selected. What is the probability that the sample proportion exceeds 0.65?

approximately normal with mean 0.60 and standard deviation 0.0310

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To find the probability that the sample proportion exceeds 0.65, we need to use the sampling distribution of sample proportions.

First, we need to verify whether the conditions for using the normal approximation to the binomial distribution are met:

1. Random Sample: The problem states that a simple random sample of 250 undergraduate students was selected. This condition is met.

2. Independence: The sample size is less than 10% of the population size. Since the population size is not given, we can assume it is large enough for this condition to be met.

3. Success-failure Condition: We need to check if both np and n(1-p) are greater than or equal to 10, where n is the sample size and p is the true proportion. In this case, np = 250 * 0.6 = 150 and n(1-p) = 250 * 0.4 = 100. Both values are greater than 10, so the condition is met.

Since the conditions are met, we can use the normal approximation to the binomial distribution. The sample proportion, p̂, follows a approximately normal distribution with mean μ = p and standard deviation σ = sqrt((p*(1-p))/n).

In this case, the population proportion, p, is 0.6 and the sample size, n, is 250. Plugging these values into the formula, we can calculate the standard deviation:

σ = sqrt((0.6 * (1-0.6))/250) ≈ 0.029

Now, we want to find the probability that the sample proportion exceeds 0.65. This can be expressed as P(p̂ > 0.65). We can convert this into a z-score using the formula:

z = (0.65 - p) / σ

Plugging in the values, we get:

z = (0.65 - 0.6) / 0.029 ≈ 1.724

Finally, we can use the standard normal distribution (Z-distribution) to find the probability associated with this z-score. By looking up the corresponding probability using a Z-table or using a calculator, we find that the probability P(p̂ > 0.65) is approximately 0.042.

Therefore, the probability that the sample proportion exceeds 0.65 is approximately 0.042, or 4.2%.