A MANUFACTURE OF LIGHTBULBS HASS SAMPLED 40 BULBS IN ORDER TO DETERMINE THE POPULATION MEAN LIFE OF THE BULBS. tHE SAMPLE MEAN WAS 1200 HOURS WITH A STANDARD DEVIATION OF 100. tHE MANUFACTURER WOULD LIKE TO KNOW, WITH 95% CONFIDENCE, THE INTERVAL VALUES FOR THE POPULATION MEAN.
To determine the interval values for the population mean with 95% confidence, we can use the concept of confidence intervals.
Step 1: Determine the sample mean and sample standard deviation.
Given:
Sample mean (x̄) = 1200 hours
Standard deviation (s) = 100
Step 2: Determine the sample size.
Given:
Sample size (n) = 40 bulbs
Step 3: Calculate the margin of error.
The margin of error (E) is determined using the formula:
E = (Z * σ) / √n
Where:
Z = Z-score corresponding to the desired level of confidence (95% confidence level corresponds to a Z-score of approximately 1.96).
σ = Population standard deviation (since the population standard deviation is unknown, we will use the sample standard deviation as an estimate, denoted as s).
n = Sample size
Substituting the values:
E = (1.96 * 100) / √40
Step 4: Calculate the confidence interval.
The confidence interval can be calculated using the formula:
Confidence interval = (x̄ - E, x̄ + E)
Substituting the values:
Confidence interval = (1200 - E, 1200 + E)
Now, let's calculate the confidence interval:
Step 5: Calculate the margin of error (E):
E = (1.96 * 100) / √40
E ≈ 27.84
Step 6: Calculate the confidence interval:
Confidence interval ≈ (1200 - 27.84, 1200 + 27.84)
Confidence interval ≈ (1172.16, 1227.84)
Therefore, with 95% confidence, the interval values for the population mean are approximately 1172.16 hours to 1227.84 hours.