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The Standard Error of the Mean (SEM) plays an important role when you want to understand the spread or variability of a sample mean relative to the true population mean. It helps us assess how precisely the sample mean estimates the actual average of the entire population.
To find the SEM, you take the known standard deviation of the population, represented as \( \sigma \), and divide it by the square root of your sample size, denoted by \( n \). Here's the formula:

• \( SEM = \frac{\sigma}{\sqrt{n}} \)

In this exercise, we know that the standard deviation \( \sigma = 25 \) hours and the sample consists of \( n = 20 \) light bulbs. Plugging these values into the formula gives us an SEM of approximately \( 5.5902 \). This computed SEM tells us how much uncertainty we should expect around the sample mean of 1014 hours due to the specific sample size and variation from the population.

Understanding a normal distribution is key when dealing with data that follows a bell-shaped curve. This type of distribution is symmetric, meaning that most of the data points fall close to the mean, with fewer data points appearing as you move away from the mean.
For example, many natural phenomena like heights, test scores, and in our case, the life in hours of light bulbs, tend to follow a normal distribution.
The characteristics of a normal distribution include:

• The mean equals the median, dividing the distribution into two equal parts.
• The curve's shape is standardized to have a peak at the mean, showing most frequent measurements.
• Empirical rule: approximately 68% of data falls within one standard deviation \( \sigma \) of the mean, 95% within two \( \sigma \), and 99.7% within three \( \sigma \).

In this problem, the normal distribution assumption allows us to use critical values like \( z_{0.025} = 1.96 \) to establish confidence intervals, stemming from the nature of the bell curve.

A 95% confidence interval offers a range that is likely to contain the true population mean with 95% certainty. It captures the uncertainty associated with sampling.To construct a two-sided confidence interval, we use the formula:

• \( \bar{x} \pm z_{\alpha/2} \times SEM \)

In our particular scenario, \( \bar{x} = 1014 \) hours, the sample mean, and \( SEM = 5.5902 \). We use a critical value of \( z_{0.025} = 1.96 \), applicable to a 95% confidence level. When plugged into the formula, this provides a confidence interval range of approximately \( (1002.0536, 1025.9464) \).
Additionally, for a one-sided 95% confidence bound (which only considers one tail of the distribution), the critical value becomes \( z_{0.05} = 1.645 \). This generates a lower confidence bound of \( 1004.8037 \) hours, noting it is slightly higher than the lower bound of the two-sided interval. This higher bound provides a more conservative estimate reflecting greater confidence that the light bulb life expectancy won't fall below this value.