According to a poll conducted in a company, 47% of the employees regularly use the internet while at work. You conduct a similar survey at your mother's office where there are 250 respondents. What is the probability you find between 102 and 117.5 employees who use the internet while at work?
Could anyone show me the steps to this problem?
To find the probability of finding between 102 and 117.5 employees who use the internet while at work, we first need to calculate the expected number of employees who fall within this range.
Step 1: Calculate the mean (expected value):
To find the mean, we multiply the total number of respondents (250) by the percentage of employees who regularly use the internet (47% or 0.47):
Mean = 250 * 0.47 = 117.5
Step 2: Calculate the standard deviation:
The standard deviation can be determined using the formula:
Standard deviation = Sqrt(N * p * (1 - p))
Where N is the number of respondents and p is the percentage of employees who regularly use the internet.
Standard deviation = Sqrt(250 * 0.47 * (1 - 0.47))
Standard deviation = Sqrt(117.85)
Step 3: Calculate the Z-scores for the lower and upper limits:
Z-score = (X - μ) / σ
Where X is the value we are interested in (102 to 117.5), μ is the mean, and σ is the standard deviation.
Z1 = (102 - 117.5) / Sqrt(117.85)
Z2 = (117.5 - 117.5) / Sqrt(117.85)
Step 4: Use a standard normal distribution table or a calculator to find the probabilities associated with the calculated Z-scores.
The probability you're looking for is equal to the difference between the probabilities of Z1 and Z2.
P(102 ≤ X ≤ 117.5) = P(Z1 ≤ Z ≤ Z2)
You can find these probabilities using a standard normal distribution table or a calculator that calculates cumulative probabilities for Z-scores.
By following these steps, you should be able to calculate the probability of finding between 102 and 117.5 employees who use the internet while at work based on the survey conducted at your mother's office.