Questions: Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A random sample of 16 fluorescent light bulbs has a mean life of 645 hours with a standard deviation of 31 hours.

Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A random sample of 16 fluorescent light bulbs has a mean life of 645 hours with a standard deviation of 31 hours.

Transcript text: Construct a 95% confidence interval for the population mean, $\mu$. Assume the population has a normal distribution. A random sample of 16 fluorescent light bulbs has a mean life of 645 hours with a standard deviation of 31 hours.

Solution

Solution Steps

Step 1: Given Data

We have a random sample of 16 fluorescent light bulbs with the following statistics:

Sample mean ($\bar{x}$) = 645 hours
Sample standard deviation ($s$) = 31 hours
Sample size ($n$) = 16
Confidence level = 95%

Step 2: Determine the Critical Value

Since the sample size is small ($n < 30$) and the population is assumed to be normally distributed, we will use the $t$-distribution. For a 95% confidence level and $n - 1 = 15$ degrees of freedom, the critical value ($t$) is approximately 2.1.

Step 3: Calculate the Standard Error

The standard error (SE) is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} = \frac{31}{\sqrt{16}} = \frac{31}{4} = 7.75 \]

Step 4: Calculate the Margin of Error

The margin of error (ME) is given by: \[ ME = t \cdot SE = 2.1 \cdot 7.75 = 16.275 \]

Step 5: Construct the Confidence Interval

The confidence interval is calculated as: \[ \text{Confidence Interval} = \bar{x} \pm ME = 645 \pm 16.275 \] This results in: \[ \text{Lower Bound} = 645 - 16.275 = 628.725 \quad \text{(rounded to 628.5)} \] \[ \text{Upper Bound} = 645 + 16.275 = 661.275 \quad \text{(rounded to 661.5)} \]