To estimate the proportion p of defective light bulbs produced in a factory, a sample of 180 bulbs were tested. In this sample 27 were defective. We will compute a 95 percent confidence interval for p. Compute the margin of error e in esimating p at 95 percent level of confidence.
To compute the margin of error (e) in estimating the proportion (p) at a 95 percent level of confidence, we can use the formula:
e = z * sqrt((p̂ * (1 - p̂)) / n),
Where:
- z is the z-score corresponding to the desired level of confidence
- p̂ is the sample proportion (27/180 in this case)
- n is the sample size (180 in this case)
First, let's calculate the sample proportion (p̂):
p̂ = Number of successes / Sample size
p̂ = 27 / 180
p̂ ≈ 0.15
Next, we need to find the value of z for a 95 percent confidence level. The z-score is determined based on the desired level of confidence using a standard normal distribution table or calculator. For a 95 percent confidence level, the z-score is approximately 1.96.
Now, we can plug in the values into the formula for the margin of error:
e = 1.96 * sqrt((0.15 * (1 - 0.15)) / 180)
Evaluating the expression:
e ≈ 0.0489
Therefore, the margin of error (e) in estimating the proportion (p) at a 95 percent level of confidence is approximately 0.0489.
Note: The margin of error represents the range within which the true population proportion is likely to fall, given the sample data and confidence level.