Estimating the Mean Age of Clientele

Solution & Explanation

To determine the 95% confidence interval for the mean age of the company's clientele, we will use the formula for a confidence interval for the mean:

$CI = \bar{x} \pm z \times \left(\frac{\sigma}{\sqrt{n}}\right)$

Where:

• $\bar{x}$ is the sample mean.
• $z$ is the z-score corresponding to the desired confidence level.
• $\sigma$ is the sample standard deviation.
• $n$ is the sample size.

Given Data:

• Sample Mean ($\bar{x}$): 35 years
• Sample Standard Deviation ($\sigma$): 5 years
• Sample Size ($n$): 100
• Confidence Level: 95%

Step-by-Step Calculation:

1. Determine the z-score for a 95% Confidence Level:

   • The z-score for a 95% confidence level is approximately 1.96. This is typically found using a standard normal distribution table or calculator.

2. Calculate the Standard Error (SE):

   • The standard error is calculated using the formula: $SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{100}} = \frac{5}{10} = 0.5$

3. Calculate the Margin of Error (ME):

   • The margin of error is calculated as: $ME = z \times SE = 1.96 \times 0.5 = 0.98$

4. Determine the Confidence Interval (CI):

   • The confidence interval is: $CI = \bar{x} \pm ME = 35 \pm 0.98$
   • Resulting in: $CI = (35 - 0.98, 35 + 0.98) = (34.02, 35.98)$

Interpretation:

The 95% confidence interval for the true mean age of the company's customers is approximately (34.02, 35.98) years. This means that if we were to take many samples, 95% of the calculated confidence intervals would contain the true mean age of the company's entire customer base. Thus, we can be 95% confident that the true average age lies within this interval based on the given sample data.