Problem 51 - Basic Statistics for Business & Economics

Question

Passenger comfort is influenced by the amount of pressurization in an airline cabin. Higher pressurization permits a closer-to-normal environment and a more relaxed flight. A study by an airline user group recorded the equivalent air pressure on 30 randomly chosen flights. The study revealed a mean equivalent air pressure of 8,000 feet with a standard deviation of 300 feet. a. Develop a \(99 \%\) confidence interval for the population mean equivalent air pressure. b. How large a sample is needed to find the population mean within 25 feet at \(95 \%\) confidence?

Step-by-Step Solution

Answer

a. The 99% confidence interval is (7,849.06, 8,150.94). b. A sample size of 555 is needed.

Step 1: Identify the Variables for Part a

We have a sample mean \( \bar{x} = 8,000 \) feet, a standard deviation \( s = 300 \) feet, and the sample size \( n = 30 \). We need a 99% confidence interval for the population mean. Determine the critical value using the t-distribution because the sample size is less than 30.

Step 2: Determine the Critical Value for Part a

For a 99% confidence interval with 29 degrees of freedom, find the critical t-value \( t^* \). Using a t-distribution table or calculator, \( t^* \approx 2.756 \).

Step 3: Calculate the Standard Error for Part a

The standard error (SE) of the mean is given by \( SE = \frac{s}{\sqrt{n}} = \frac{300}{\sqrt{30}} \approx 54.77 \).

Step 4: Construct the Confidence Interval for Part a

The confidence interval is given by \( \bar{x} \pm t^* \cdot SE \). Thus, \( 8,000 \pm 2.756 \times 54.77 \). Calculate the interval: \( 8,000 \pm 150.94 \) gives \( (7,849.06, 8,150.94) \).

Step 5: Identify the Variables for Part b

We need to determine the sample size \( n \) necessary to estimate the population mean within 25 feet at a 95% confidence level. Here, the margin of error \( E = 25 \), and the standard deviation \( s = 300 \).

Step 6: Determine the Critical Value for Part b

For a 95% confidence level, the z-score (because it's a large sample requirement) is \( z^* \approx 1.96 \).

Step 7: Calculate the Required Sample Size for Part b

The formula for sample size \( n \) is \( n = \left(\frac{z^* \cdot s}{E}\right)^2 \). Substitute the values \( n = \left(\frac{1.96 \times 300}{25}\right)^2 \approx 554.88 \). Rounding up, \( n = 555 \).

Key Concepts

T-distribution

The t-distribution is essential in statistics, especially when dealing with small sample sizes. It is a type of probability distribution that's similar to the normal distribution but has thicker tails. This means there is more probability in the extreme ends of the distribution, which allows for more uncertainty. A t-distribution is used when the sample size is below 30, offering a more accurate reflection of variability than the normal distribution.

• It is vital in constructing confidence intervals when the sample is small and the population standard deviation is not known.
• The t-distribution is characterized by "degrees of freedom", which depend on the sample size (n-1 for a sample of size n).
• As the sample size increases, the t-distribution approaches the shape of a normal distribution.

In our exercise, since the sample size is 30, the t-distribution is more appropriate for finding the critical value needed to construct the confidence interval.

Standard Error

Standard error (SE) is a crucial statistic that reflects the variability of a sample mean estimation. It essentially measures how accurately the sample mean represents the population mean. The formula for SE is:\[SE = \frac{s}{\sqrt{n}}\]where \(s\) is the sample standard deviation, and \(n\) is the sample size. A smaller standard error means the sample mean is a more accurate reflection of the population mean.

• Standard error decreases as the sample size increases, implying more reliable sample estimates.
• A smaller SE leads to a narrower confidence interval, indicating greater precision.

For example, in the airline study, with a sample size of 30 and a standard deviation of 300, the SE was calculated as approximately 54.77. This SE was used to construct the 99% confidence interval for the mean equivalent air pressure.

Sample Size Determination

Determining the appropriate sample size is crucial for statistical accuracy and cost-efficiency in any study. Larger sample sizes result in more reliable estimates but can be more expensive and time-consuming. The sample size is often selected to ensure a specific margin of error at a particular confidence level.

For a desired margin of error (E), the formula to calculate required sample size (n) is:\[n = \left( \frac{z^* \cdot s}{E} \right)^2\]In this formula:

• \(z^*\) is the z-score corresponding to the desired confidence level (e.g., 1.96 for 95%).
• \(s\) is the sample standard deviation.
• \(E\) is the maximum acceptable margin of error.

In part b of our original exercise, the aim was to find the sample size needed to estimate the population mean within 25 feet with 95% confidence. With a given standard deviation of 300 and a z-score of 1.96, the sample size calculated was approximately 555.