30 percent of the employees at a large biotech firm are working from home. [You may find it useful to reference the z table.]
a. In a sample of 40 employees, what is the probability that more than 20% of them are working from home? (Round final answer to 4 decimal places.)
b. In a sample of 280 employees, what is the probability that more than 20% of them are working from home? (Round final answer to 4 decimal places.)
c. Comment on the reason for the difference between the computed probabilities in parts a and b.
multiple choice
As the sample number increases, the probability of more than 20% also increases, due to the lower z value and decreased standard error.
As the sample number increases, the probability of more than 20% also increases, due to the lower z value and increased standard error.
The correct answer is: As the sample number increases, the probability of more than 20% also increases, due to the lower z value and decreased standard error.
To answer these questions, we need to use the normal distribution and the z-score. The formula for finding the z-score is:
z = (x - μ) / (σ / √n)
Where:
- z is the z-score
- x is the sample proportion
- μ is the population proportion
- σ is the standard deviation
- n is the sample size
Given that 30% of the employees are working from home, we can assume that the population proportion (μ) is 0.30. We also need to calculate the standard deviation (σ) using the formula:
σ = √(μ * (1 - μ) / n)
Now we can proceed to calculate the probabilities.
a. In a sample of 40 employees:
The sample size (n) is 40. To find the probability that more than 20% of them are working from home, we need to calculate the z-score for 0.20:
z = (0.20 - 0.30) / (√(0.30 * (1 - 0.30) / 40))
Using a Z-table, we can find the probability associated with this z-score.
b. In a sample of 280 employees:
The sample size (n) is 280. Again, we need to calculate the z-score for 0.20:
z = (0.20 - 0.30) / (√(0.30 * (1 - 0.30) / 280))
Using a Z-table, we can find the probability associated with this z-score.
c. Comment on the reason for the difference between the computed probabilities in parts a and b:
To answer this question, we need to compare the calculated z-scores in both parts. The z-score is influenced by the sample size (n). As the sample size increases, the standard error decreases, resulting in a smaller z-score. This means that the probability of more than 20% of the employees working from home will increase. Therefore, the correct choice would be: "As the sample number increases, the probability of more than 20% also increases, due to the lower z value and increased standard error."
To find the probabilities in both parts a and b, we'll need to use the z-table, which provides the area under the standard normal curve.
Let's break down the process step by step:
a. To find the probability that more than 20% (0.2) of the employees are working from home in a sample of 40 employees, we need to calculate the z-score and then use the z-table.
Step 1: Calculate the sample proportion:
Sample proportion (p̂) = 0.3 (since 30% of employees are working from home)
Step 2: Calculate the standard deviation of the sample proportion:
Standard deviation (σ) = √[(p̂ * (1 - p̂)) / n]
Here, n represents the sample size.
σ = √[(0.3 * (1 - 0.3)) / 40]
σ = √[0.21 / 40]
σ ≈ 0.0912871
Step 3: Calculate the z-score:
z = (x - μ) / σ
Here, x represents the value we are interested in (0.2), and μ represents the population mean (0.3).
z = (0.2 - 0.3) / 0.0912871
z ≈ -1.0954
Step 4: Look up the z-score in the z-table to find the corresponding probability. Since we're interested in the probability of more than 20% (0.2), we need to find the area under the curve to the left of -1.0954 and then subtract it from 1.
Using the z-table or a statistical software, you can find that the area to the left of -1.0954 is approximately 0.1368.
P(X > 0.2) = 1 - 0.1368 ≈ 0.8632
Therefore, the probability that more than 20% of the sample of 40 employees are working from home is approximately 0.8632.
b. Now, let's find the probability for a sample of 280 employees.
Step 1: Calculate the sample proportion (p̂) and standard deviation (σ) using the same formulas as before. Only the sample size (n) changes to 280.
p̂ = 0.3 (same as before)
σ = √[(0.3 * (1 - 0.3)) / 280]
σ ≈ 0.030878
Step 2: Calculate the z-score:
z = (x - μ) / σ
Again, x is 0.2, and μ is 0.3.
z = (0.2 - 0.3) / 0.030878
z ≈ -3.234
Step 3: Look up the z-score in the z-table to find the corresponding probability. Since we're interested in the probability of more than 20% (0.2), we need to find the area under the curve to the left of -3.234 and then subtract it from 1.
Using the z-table or a statistical software, you can find that the area to the left of -3.234 is approximately 0.0006.
P(X > 0.2) = 1 - 0.0006 ≈ 0.9994
Therefore, the probability that more than 20% of the sample of 280 employees are working from home is approximately 0.9994.
c. Now, let's analyze the reason for the difference between the computed probabilities in parts a and b.
As the sample number increases, the probability of more than 20% also increases due to the lower z-value and increased standard error. The z-value is calculated using the standard deviation of the sample proportion, which decreases as the sample size increases. Consequently, the z-value becomes more negative, resulting in a larger area under the curve to the left (lower probability). This means that as the sample size gets larger, the probability of more than 20% decreases because the estimation becomes more precise with larger samples.