Problem 55 - Basic Statistics for Business and Economics with Student

Question

55\. 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 corresponding air pressure on 30 randomly chosen flights. The study revealed a mean equivalent pressure of 8,000 feet with a standard deviation of 300 feet. a. Develop a 99 percent confidence interval for the population mean equivalent pressure. b. How large a sample is needed to find the population mean within 25 feet at 95 percent confidence?

Step-by-Step Solution

Answer

a. CI: [7858.829, 8141.171] feet; b. Sample size: 554 flights.

Step 1: Determine the Critical Value for Part (a)

To find a 99% confidence interval for the population mean, we need to determine the critical value (z-score) that corresponds to the desired confidence level. For a 99% confidence level, the critical value is approximately 2.576. This value is obtained from a standard normal distribution table.

Step 2: Calculate the Margin of Error for Part (a)

The margin of error (ME) is calculated using the formula: \( ME = z \times \frac{s}{\sqrt{n}} \), where \( z \) is the critical value, \( s \) is the standard deviation, and \( n \) is the sample size. Substituting the values, we get:\[ ME = 2.576 \times \frac{300}{\sqrt{30}} \approx 141.171 \] feet.

Step 3: Construct the Confidence Interval for Part (a)

The confidence interval is given by the formula: \( \text{CI} = \bar{x} \pm ME \), where \( \bar{x} \) is the sample mean. Given \( \bar{x} = 8000 \) feet, the confidence interval is:\[ 8000 \pm 141.171 \] This results in an interval of \( [7858.829, 8141.171] \) feet.

Step 4: Determine Required Sample Size for Part (b)

To find the sample size needed for a specified margin of error, use the formula: \( n = \left( \frac{z \times s}{E} \right)^2 \), where \( E \) is the desired margin of error. For 95% confidence, the critical z-value is 1.96. With \( E = 25 \) feet, we have:\[ n = \left( \frac{1.96 \times 300}{25} \right)^2 = \left( 23.52 \right)^2 = 553.1904 \]This rounds up to 554 because you cannot have a fraction of a flight.

Key Concepts

Sample Size Calculation

Understanding how to calculate the right sample size is crucial for obtaining reliable data in a study. It ensures that the estimates we develop are precise and representative of the entire population.
When calculating sample size, we want to determine how many observations we need to confidently make statements about the population from our sample.

For a specific desired margin of error (ME), which represents the maximum expected difference between the true population parameter and the sample estimate, the formula to calculate sample size is:

• \( n = \left( \frac{z \times s}{E} \right)^2 \)
• Here, \( n \) is the sample size, \( z \) is the critical value associated with the confidence level, \( s \) is the standard deviation, and \( E \) is the margin of error.

In the airline pressurization example, the critical value for 95% confidence is 1.96, and when we want the population mean within 25 feet, the calculation gives a sample size of about 554 flights.
This figure is rounded up as you cannot sample a fraction of a flight.

Margin of Error

The margin of error (ME) plays a significant role in determining the accuracy of our confidence intervals. It tells us how much we can expect our sample's results to vary from the true population parameter.
In simpler terms, it allows us to account for the possibility of sampling error.

To calculate the margin of error, we use the formula:

• \( ME = z \times \frac{s}{\sqrt{n}} \)
• where \( z \) is the critical value, \( s \) is the sample standard deviation, and \( n \) is the sample size.

If we have a larger sample, our margin of error decreases because the sample provides a better approximation of the population.
In the airline example, choosing an appropriate margin of error was crucial for ensuring precise estimates of the average cabin pressure.

Critical Value

Critical values are fundamental in the construction of confidence intervals. They represent the number of standard deviations a point is from the mean in a standard normal distribution.
To find these values, we refer to Z-tables or T-tables, depending on the sample size and variability known.

• The critical value \( z \) directly influences the width of the confidence interval.
• For a 99% confidence level, the critical value is approximately 2.576.
• This means we expect the true mean to fall within 2.576 standard deviations of our sample mean in 99 out of 100 similar samples.

In leveling cabin pressure, obtaining the correct critical value allowed for a meaningful interval to be calculated, signifying very high confidence in the results.