Question

In an effort to estimate the mean amount spent per customer for dinner at a major Atlanta restaurant, data were collected for a sample of 49 customers. Assume a population standard deviation of \$5. a. At \(95 \%\) confidence, what is the margin of error? b. If the sample mean is \(\$ 24.80\), what is the \(95 \%\) confidence interval for the population mean?

Short answer

The margin of error for a 95% confidence level is approximately \$1.40. The 95% confidence interval for the population mean amount spent per customer is approximately between \$23.40 and \$26.20.

Step-by-step solution

Identifying the given values

We are given the following values: Sample size (n) = 49 customers Population standard deviation (σ) = \$5 Sample mean (\(\bar{x}\)) = \$24.80 Confidence level = 95%

Calculating the Margin of Error

To calculate the margin of error, we need to determine the value of the Standard Error. The Standard Error is given by the formula SE = \(\frac{\sigma}{\sqrt{n}}\). Then, we will use the Z-score of 1.96 for a 95% confidence level. The formula for margin of error (ME) is: ME = Z * SE First, calculate the Standard Error (SE): SE = \(\frac{\sigma}{\sqrt{n}}\) SE = \(\frac{5}{\sqrt{49}}\) SE ≈ 0.714 Now, calculate the Margin of Error (ME): ME = Z * SE ME = 1.96 * 0.714 ME ≈ 1.40 The margin of error for a 95% confidence level is approximately \$1.40.

Constructing the Confidence Interval

Now that we have the margin of error, we can construct the confidence interval for the population mean using the following formula: CI = \(\bar{x} \pm \) ME CI = \$24.80 ± \$1.40 This gives us the 95% confidence interval: CI = (\$23.40, \$26.20) The 95% confidence interval for the population mean amount spent per customer is approximately between \$23.40 and \$26.20.

Key concepts

Margin of Error

The margin of error is a crucial component when constructing a confidence interval, as it captures the uncertainty or potential variance within your data. It's essentially an allowance for error in your statistical estimate, indicating the range within which the true population parameter is likely to lie.
In a practical sense, the margin of error tells us how "off" our sample mean (\( \bar{x} \)) might be from the true population mean. When we calculate the margin of error, we rely on the level of confidence we wish to have – in this case, 95%.
This is why a Z-score (in this context, a widely accepted standard of 1.96 for 95% confidence) is used. The formula for calculating the margin of error is:

• ME = Z \( \times \) SE

where ME is the margin of error, Z is the Z-score, and SE is the standard error.
In the problem, we calculated the margin of error as approximately \(\\(1.40\), meaning that the actual average amount spent per customer is likely to be within \(\\)1.40\) of our sample mean of \(\$24.80\). This gives us the assurance that our estimate is statistically sound.

Sample Size

The sample size is a vital factor in determining the precision of statistical estimates. In our exercise, the sample size (\( n \)) was 49 customers, a number that significantly influences the standard error and margin of error.
The sample size affects the width of the confidence interval you calculate; larger samples tend to yield more precise estimates because they reduce the standard error.
Here's why a bigger sample size matters:

• It reduces the margin of error, making your confidence interval narrower and thus your estimate more precise.
• A larger sample size generally leads to more reliable and valid results.

In the formula for the standard error, \[SE = \frac{\sigma}{\sqrt{n}}\] the sample size is under the square root in the denominator:

• The larger the sample size \( n \), the smaller the standard error \( SE \).
• With a smaller SE, our margin of error also decreases, reflecting our improved estimate's reliability.

Therefore, increasing the sample size will lead to a tighter confidence interval with a lower margin of error.

Standard Error

When dealing with sample data, it's essential to understand the concept of standard error, as it quantifies the variability or spread of the sample mean estimates around the true population mean. The standard error is an estimation of how much the sample mean (\( \bar{x} \)) would vary from one sample to another.
In our exercise, the standard error (\( SE \)) was calculated using the formula: \[SE = \frac{\sigma}{\sqrt{n}}\]Here, \( \sigma \) represents the population standard deviation, and \( n \) is the sample size.
For our scenario,

• Population standard deviation \( \sigma \)= \( \\(5\)
• Sample size \( n \)= 49
• SE = \( \frac{5}{\sqrt{49}} = \\)0.714 \)

The smaller the standard error, the more accurately the sample mean represents the population mean.
A lower standard error suggests that there's less "noise" affecting the precision of your estimate, leading to a tighter confidence interval. This makes your data analysis more robust and statistically reliable. For this reason, minimizing the standard error through means such as increasing sample size can greatly enhance the quality of statistical inference.