Question

A random sample of 260 workers in a high-rise office building revealed that \(30 \%\) were very satisfied with the quality of elevator service. At the \(99 \%\) level, what is your estimate of the population value?

Short answer

The 99% confidence interval estimate of the population value is [0.2269, 0.3731].

Step-by-step solution

Identify Variables and Given Data

Given: Sample size ( = 260, Sample proportion (p̂) = 30% or 0.30, Confidence level = 99%. Standard confidence z-value for 99% confidence level is 2.576.

Calculate Standard Error

The formula for standard error (SE) of the sample proportion is: \[ SE = \sqrt{\frac{p\(1-p\)}{n}} \] Substituting the values, \[ SE = \sqrt{0.30\(1 - 0.30\)}{260} \] = 0.0284.

Determine Margin of Error

The margin of error (ME) is calculated using the z-value and the standard error. \[ ME = z \cdot SE \] Substituting the values, \[ ME = 2.576 \cdot 0.0284 = 0.0731 \].

Calculate Confidence Interval

The 99\replaceable level confidence interval for the population proportion is: \[ p\replaceable \pm ME \] Substituting the values, the interval is \[ 0.30 \pm 0.0731 \]. Therefore, the interval is [0.2269, 0.3731].

Key concepts

Sample Size

The sample size is quite simply the number of observations or measurements taken from the overall population. In our example, we have a sample size of 260 workers. A larger sample size tends to result in more reliable and accurate estimates because it better represents the population.

To decide on a sample size, consider factors like:

• The desired confidence level
• The acceptable margin of error
• The population size and variability

Remember, a good sample is random and representative, reducing biases and improving the study's accuracy.

Sampling Proportion

Sampling proportion is the percentage or fraction of the sample that has a particular characteristic. In our example, 30% of the 260 sampled workers are very satisfied with the elevator service. This value is called the sample proportion and is represented by \(\hat{p}\).

It's calculated as: \(\hat{p} = \frac{x}{n}\) where:

• \(x\) = number of favorable outcomes (78 workers very satisfied)
• \(n\) = total sample size (260)

Example calculation: \(0.30 = \frac{78}{260}\).
Our goal is to estimate the population proportion from this sample proportion.

Standard Error

Standard Error (SE) indicates the accuracy of a sample proportion in estimating the population proportion. It shows how much the sample proportion \(\hat{p}\) is expected to vary from the true population proportion \(p\). The formula for SE is:
\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
In our example,
\[ SE = \sqrt{\frac{0.30(1-0.30)}{260}} = \sqrt{\frac{0.21}{260}} = 0.0284 \]

Lower SE values indicate more precise estimates. Larger samples usually result in smaller SEs.

Margin of Error

Margin of Error (ME) reflects the range within which the true population parameter is expected to fall. It combines the SE with a critical value (z-value) based on the desired confidence level. The formula is:
\[ ME = z \times SE \]
For a 99% confidence level, the z-value is approximately 2.576. In our example,
\[ ME = 2.576 \times 0.0284 = 0.0731 \]
Margin of Error helps to create a confidence interval for the estimate.

Confidence Level

The confidence level indicates the degree of certainty that the population parameter lies within the confidence interval. Common confidence levels are 90%, 95%, and 99%. In our example, we use a 99% confidence level, meaning we are 99% confident the interval contains the true population proportion.

To compute the confidence interval, we use: \[ \hat{p} \pm ME \]
Substituting in our values:
\[ 0.30 \pm 0.0731 = (0.2269, 0.3731) \]
This interval tells us that between 22.69% and 37.31% of the entire population of workers are very satisfied with the elevator service.