At a certain university, 50% of all entering freshmen planned to major in a STEM (science, technology, engineering, mathematics) discipline. A sample of 36 freshmen is selected. What is the probability that the proportion of freshmen in the sample is between 0.475 and 0.580?
Write only a number as your answer. Round to 4 decimal places (for example 0.3748). Do not write as a percentage.
i just need to know how to do it
To find the probability, we need to calculate the sampling distribution of proportions for a sample size of 36.
The sampling distribution of proportions follows a normal distribution when the sample size is large enough (n * p > 5 and n * (1-p) > 5). In this case, the sample size is 36 and the estimated proportion is 0.50.
To calculate the standard deviation of the sampling distribution, we use the formula:
Standard Deviation (SD) = sqrt(p * (1-p) / n)
where p is the estimated proportion and n is the sample size.
So, SD = sqrt(0.50 * (1-0.50) / 36) = 0.0816
To find the z-scores for the lower and upper bounds of the proportion range (0.475 and 0.580), we use the formula:
z = (p - mean) / SD
where p is the proportion, mean is the expected proportion, and SD is the standard deviation.
For the lower bound:
z1 = (0.475 - 0.50) / 0.0816 = -0.3053
For the upper bound:
z2 = (0.580 - 0.50) / 0.0816 = 0.9804
Next, we need to find the cumulative probabilities of these z-scores using a standard normal distribution table or a calculator.
Using a standard normal distribution table or a calculator, the cumulative probability for z1 (-0.3053) is 0.3801, and the cumulative probability for z2 (0.9804) is 0.8365.
To find the probability between the lower and upper bounds, we subtract the cumulative probability for the lower bound from the cumulative probability for the upper bound:
Probability = 0.8365 - 0.3801 = 0.4564
Rounded to 4 decimal places, the probability that the proportion of freshmen in the sample is between 0.475 and 0.580 is 0.4564.
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