In a local university,40% of the students live in the dormitories. A random sample of 80 students is selected for a particular study. What is the probability that the sample proportion (the proportion living in the dormitories) is between 0.30 and 0.50?
n a local university,40% of the students live in the dormitories. A random sample of 80 students is selected for a particular study. What is the probability that the sample proportion (the proportion living in the dormitories) is between 0.30 and 0.50?
To find the probability that the sample proportion is between 0.30 and 0.50, we need to calculate the z-scores for both boundaries and then find the area under the normal curve between those z-scores.
Step 1: Calculate the mean and standard deviation.
Given:
Total sample size (n) = 80
Proportion of students living in dormitories (p) = 0.40
Mean (μ) = p = 0.40
Standard Deviation (σ) = √[(p * (1 - p))/n]
= √[(0.40 * (1 - 0.40))/80]
= √[(0.24)/80]
= √0.003
≈ 0.055
Step 2: Calculate the z-scores for both boundaries.
z1 = (x1 - μ) / σ
= (0.30 - 0.40) / 0.055
≈ -1.82
z2 = (x2 - μ) / σ
= (0.50 - 0.40) / 0.055
≈ 1.82
Step 3: Find the area under the normal curve between these z-scores.
Using a standard normal distribution table or a calculator, the area to the left of z = -1.82 is approximately 0.0344, and the area to the left of z = 1.82 is also approximately 0.9656.
Therefore, the probability that the sample proportion is between 0.30 and 0.50 is:
= 0.9656 - 0.0344
= 0.9312
≈ 93.12%
To find the probability that the sample proportion is between 0.30 and 0.50, we can use the normal approximation to the binomial distribution.
First, we need to calculate the mean and standard deviation of the sample proportion.
The mean of the sample proportion (denoted by p̂) can be calculated by multiplying the proportion of students living in the dormitories (p) by the total number of students in the sample (n):
p̂ = p * n
In this case, p = 0.40 (proportion of students living in the dormitories) and n = 80 (sample size), so:
p̂ = 0.40 * 80 = 32
The standard deviation of the sample proportion (denoted by σ̂) can be calculated using the following formula:
σ̂ = √(p * (1 - p) / n)
In this case:
σ̂ = √(0.40 * (1 - 0.40) / 80) = √(0.24 / 80) = √0.003 = 0.055
Now that we have the mean and standard deviation of the sample proportion, we can use the normal distribution to calculate the probability that the sample proportion is between 0.30 and 0.50.
We can standardize the values of 0.30 and 0.50 by subtracting the mean (p̂) and dividing by the standard deviation (σ̂):
Standardized value for 0.30: (0.30 - 0.40) / 0.055 = -1.82
Standardized value for 0.50: (0.50 - 0.40) / 0.055 = 1.82
Now, we can look up the corresponding probabilities in the standard normal distribution table, or use software/tools like Excel, R, or Python to find the area under the curve between these standardized values.
The probability that the sample proportion is between 0.30 and 0.50 is the difference between the cumulative probabilities at the two standardized values:
P(0.30 ≤ p̂ ≤ 0.50) = P(Z ≤ 1.82) - P(Z ≤ -1.82)
You can use the standard normal distribution table or a statistical software/tool to find these probabilities.