The new Twinkle bulb is being developed to last more than 1000 hours. A random sample of 100 of these new bulbs is selected from the production line. It was found that 48 lasted more than 1000 hours. Find the point estimate for the population proportion, the margin of error for a 95% confidence and then construct to the 95% confidence interval for the population proportion p.
What is the margin of error E for the 95% confidence interval? Round E to three decimal places.
To find the margin of error (E) for a 95% confidence interval for the population proportion (p), you can use the following formula:
E = z * sqrt((p̂ * q̂) / n)
Where:
- E is the margin of error
- z is the z-score corresponding to the desired confidence level (in this case, 95%)
- p̂ is the point estimate for the population proportion
- q̂ is (1 - p̂), the complementary probability of p̂
- n is the sample size
First, let's find the point estimate for the population proportion (p̂) based on the given information. The point estimate is simply the proportion of the sample that meets the criterion (i.e., lasting more than 1000 hours). In this case, it is 48 out of 100 bulbs:
p̂ = 48 / 100 = 0.48
Next, calculate q̂ as (1 - p̂):
q̂ = 1 - 0.48 = 0.52
Now, you need to find the z-score corresponding to a 95% confidence level. The z-score can be obtained using statistical tables or a calculator. For a 95% confidence level, the z-score is approximately 1.96.
Substituting the values into the formula, we have:
E = 1.96 * sqrt((0.48 * 0.52) / 100)
Calculating the square root:
E = 1.96 * sqrt(0.2496 / 100)
Simplifying:
E = 1.96 * 0.04996
E ≈ 0.0979 (rounded to three decimal places)
Therefore, the margin of error (E) for the 95% confidence interval is approximately 0.098.