Problem 37 - Algebra 2

Question

In a poll of 2750 airline travelers, 138 said they never check their luggage when they fly. Find the sample proportion, the margin of error, and the interval likely to contain the true population proportion.

Step-by-Step Solution

Answer

The sample proportion is 0.05018, the margin of error is 0.0143, and the interval likely to contain the true population proportion is [0.03588, 0.06448].

Step 1: Calculate Sample Proportion

The sample proportion \(p\) is calculated as the number of successful outcomes divided by the total number of outcomes. In this context, a successful outcome is a traveler who never checks their luggage when they fly. Thus, \(p = 138 / 2750 = 0.05018\).

Step 2: Calculate Margin of Error

The margin of error (E) for a proportion is computed as \[E = Z * \sqrt{ \frac{p * (1 - p)} {n} }\] where \(Z = 1.96\) (the z-score which corresponds to a 95% confidence level), \(p = 0.05018\) (the sample proportion), and \(n = 2750\) (the sample size). Substituting these values in, we find that \(E = 1.96 * \sqrt{ \frac{0.05018 * (1 - 0.05018)} {2750} } = 0.0143\).

Step 3: Calculate Confidence Interval

The confidence interval is then constructed as \(p ± E\), which in this case is \(0.05018 ± 0.0143\), gives an interval of [0.03588, 0.06448]. This is the interval likely to contain the true population proportion.

Key Concepts

Sample Proportion

The concept of sample proportion is vital in statistics, especially when trying to estimate the characteristics of a large population based on a smaller, more manageable group, known as a sample. The sample proportion is a ratio that represents the fraction of the sample that exhibits a certain trait or attribute. In our exercise, this trait is airline travellers who never check their luggage. To calculate the sample proportion, you divide the number of occurrences of the trait (138 in our case) by the total sample size (2750). This gives us \[p = \frac{138}{2750} = 0.05018\]This value, 0.05018, means that approximately 5% of the sampled airline travelers in this poll say they never check their luggage. Understanding sample proportion is crucial as it serves as a foundational component for further statistical analyses, such as calculating confidence intervals.

Confidence Interval

Confidence intervals provide a range of values which are believed to encompass the true value of a population parameter. When estimating a population proportion, the confidence interval helps us assess the accuracy of our sample proportion.

• The confidence level, commonly set at 95%, indicates that if the same population survey were conducted repeatedly, approximately 95% of the calculated intervals would include the true population proportion.
• The margin of error is added and subtracted from the sample proportion to create the interval boundary.

For our sampled airline travelers, the confidence interval is \[0.05018 \pm 0.0143\]This results in a range from 0.03588 to 0.06448. Thus, we are 95% confident that the true proportion of all airline travelers who never check their luggage lies between 3.588% and 6.448%.

Population Proportion

The population proportion is the actual fraction of the entire population that possesses a particular attribute or characteristic. Unlike the sample proportion, which is derived from a subset of the population, the population proportion is a fixed but often unknown value.

• It's what researchers aim to estimate through surveys, polls, or experiments.
• The true population proportion might be outside the sample proportion's confidence interval due to random sampling error, but this is rare when using a large and representative sample.

In the context of our example, the true population proportion of airline travelers who never check their luggage is what we are estimating. By using the sample proportion and calculating a confidence interval, we make an educated guess about this unknown value. Thus, even though the population proportion remains unseen, statistical tools allow researchers to make reasonable and reliable estimates based on observed data.