Forty-five percent of a large class of students in Statistics are females. Suppose 28 students are selected at random from the class. Use a normal approximation with continuity correction to calculate the probability that the number of males in the sample is at least 12 but not more than 18.
To calculate the probability that the number of males in the sample is between 12 and 18 (inclusive), we need to use the normal approximation with continuity correction.
Step 1: Convert the number of males to a standard normal variable.
First, calculate the expected number of males in the sample:
Expected number of males = (1 - 0.45) * 28 = 0.55 * 28 = 15.4
Then, calculate the standard deviation of the number of males:
Standard deviation = sqrt(28 * 0.45 * 0.55) = sqrt(6.615) ≈ 2.57
Next, calculate the standard deviation of the sample proportion:
Standard deviation (proportion) = sqrt((0.45 * 0.55) / 28) ≈ sqrt(0.00845) ≈ 0.092
Step 2: Apply the continuity correction.
To apply the continuity correction, we subtract 0.5 from each endpoint:
Lower endpoint: 12 - 0.5 = 11.5
Upper endpoint: 18 + 0.5 = 18.5
Step 3: Standardize the endpoints.
To standardize the endpoints, subtract the expected value and divide by the standard deviation:
Standardized lower endpoint: (11.5 - 15.4) / 2.57 ≈ -1.52
Standardized upper endpoint: (18.5 - 15.4) / 2.57 ≈ 1.20
Step 4: Calculate the probabilities.
Using the standard normal distribution, calculate the probability between the standardized endpoints:
Probability = P(-1.52 ≤ Z ≤ 1.20)
You can use a table or calculator to find this probability. For example, using a standard normal distribution calculator, we find that P(-1.52 ≤ Z ≤ 1.20) is approximately 0.5582.
Therefore, the probability that the number of males in the sample is at least 12 but not more than 18, using a normal approximation with continuity correction, is approximately 0.5582.
To calculate the probability that the number of males in the sample is at least 12 but not more than 18, we need to use a normal approximation with continuity correction.
Here's how you can calculate it step by step:
Step 1: Calculate the mean and standard deviation of the distribution.
The mean (μ) of the number of males in the sample can be calculated as follows:
μ = n*p = 28 * (1 - 0.45) = 28 * 0.55 = 15.4
The standard deviation (σ) of the number of males in the sample can be calculated using the formula:
σ = sqrt(n*p*(1-p)) = sqrt(28 * 0.45 * (1 - 0.45)) ≈ 3.33
Step 2: Apply the continuity correction.
To account for the continuity correction, we adjust the boundaries of the desired range by adding or subtracting 0.5 from each value. So, we need to calculate the probability that the number of males is between 11.5 and 18.5.
Step 3: Standardize the values.
We can standardize the values using the standard normal distribution (z-score) formula:
z = (x - μ) / σ
For the lower boundary:
z1 = (11.5 - 15.4) / 3.33 ≈ -1.18
For the upper boundary:
z2 = (18.5 - 15.4) / 3.33 ≈ 0.93
Step 4: Look up the probabilities.
Now, we can look up the probabilities corresponding to the standardized values in the standard normal distribution table. We want to find the probability that the number of males is between -1.18 and 0.93.
From the standard normal distribution table, we find the following probabilities:
P(z ≤ -1.18) = 0.119
P(z ≤ 0.93) = 0.823
Step 5: Calculate the desired probability.
We want to find the probability that the number of males is at least 12 but not more than 18. We can calculate this by subtracting the probability of being less than 12 from the probability of being less than or equal to 18.
P(12 ≤ X ≤ 18) = P(X ≤ 18) - P(X < 12)
= P(z ≤ 0.93) - P(z < -1.18)
= 0.823 - 0.119
= 0.704
Therefore, the probability that the number of males in the sample is at least 12 but not more than 18 is approximately 0.704.