A manufacturer of light bulbs has sampled 40 bulbs in order to determine the population mean life of the bulbs.
The sample mean was 1,200 hours with a standard deviation of 100. The manufacturer would like to know, with
95% confidence, the interval values for the population mean
95% = mean ± 1.96 SEm
SEm = SD/√n
To calculate the interval values for the population mean with 95% confidence, we can use the concept of the confidence interval.
The formula for calculating a confidence interval for the population mean is:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Now, let's break down the steps to calculate the interval values:
Step 1: Determine the critical value.
The critical value is based on the desired confidence level and the sample size. Since you want a 95% confidence level, we need to find the critical value for a 95% confidence interval. This critical value corresponds to the z-score, which can be obtained from the standard normal distribution table or a statistical calculator. For a 95% confidence level, the critical value is approximately 1.96.
Step 2: Calculate the standard error.
The standard error measures the variability in the sample mean. It can be calculated using the formula: Standard Error = Standard Deviation / √(Sample Size). In this case, the standard deviation given is 100, and the sample size is 40, so the standard error would be 100 / √(40) ≈ 15.81.
Step 3: Calculate the confidence interval.
Using the formula mentioned earlier, we can plug in the values to calculate the confidence interval:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Confidence Interval = 1200 ± (1.96 * 15.81)
Calculating this, we get:
Confidence Interval = 1200 ± 30.98
So, with 95% confidence, the interval values for the population mean would be approximately (1169.02, 1230.98).