74% of freshmen entering public high schools in 2006 graduated with their class in 2010. A random sample of 81 freshmen is selected. Find the probability that the proportion of students who graduated is greater than 0.750 .
Write only a number as your answer. Round to 4 decimal places (for example 0.1048). Do not write as a percentage.
To find the probability that the proportion of students who graduated is greater than 0.750, we need to use the concept of sampling distributions and the Central Limit Theorem.
First, let's calculate the mean and standard deviation of the sampling distribution using the given information. In this case, the mean (μ) is equal to the proportion of freshmen who graduated, which is 0.74. The standard deviation (σ) can be calculated using the formula:
σ = sqrt[(p * q) / n]
where p is the proportion of freshmen who graduated (0.74), q is 1 - p (0.26), and n is the sample size (81 in this case).
σ = sqrt[(0.74 * 0.26) / 81]
Using a calculator or spreadsheet, we can find that σ ≈ 0.0440.
Now, we need to standardize the proportion of students who graduated (0.750) using the formula:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean of the sampling distribution (0.74), and σ is the standard deviation of the sampling distribution (0.0440).
z = (0.750 - 0.74) / 0.0440
Calculating this, we get z ≈ 0.2273.
Finally, we can use a standard normal distribution table or a calculator to find the probability of getting a proportion greater than 0.750 (which corresponds to a z-value greater than 0.2273).
Looking up the z-value of 0.2273 in the standard normal distribution table, we find that the probability is approximately 0.5899.
Therefore, the probability that the proportion of students who graduated is greater than 0.750 is approximately 0.5899.