A company has developed a new type of light bulb, and wants to estimate its mean lifetime. A simple random sample of 12 bulbs had a sample mean lifetime of 692 hours with a sample standard deviation of 30 hours. It is reasonable to believe that the population is approximately normal. Find the lower bound of the 95% confidence interval for the population mean lifetime of all bulbs manufactured by this new process.
Round to the nearest integer. Write only a number as your answer. Do not write any units.
692-+ 1.96* 30 /sqrt(12)
692 -+ 16.97
(675, 709)
To find the lower bound of the 95% confidence interval for the population mean lifetime of the light bulbs, we can use the formula:
Lower bound = sample mean - (critical value) * (sample standard deviation / sqrt(sample size))
First, we need to find the critical value corresponding to a 95% confidence level. Since the population is believed to be approximately normal, we can use the t-distribution for small sample sizes. With a sample size of 12 - 1 = 11, we can find the critical value from a t-table or calculator.
For a 95% confidence level and 11 degrees of freedom, the critical value is approximately 2.201.
Next, we can substitute the values into the formula:
Lower bound = 692 - (2.201) * (30 / sqrt(12))
Lower bound ≈ 692 - (2.201) * (30 / 3.464)
Lower bound ≈ 692 - (2.201) * 8.660
Lower bound ≈ 692 - 19.072
Lower bound ≈ 672
Rounded to the nearest integer, the lower bound of the 95% confidence interval for the population mean lifetime of the bulbs is 672.
Answer: 672