Questions: According to the Transportation Security Administration (TSA) data, 9% of the passengers fly first-class. A random sample of size 44 was obtained. Let p̂ be the proportion of the sample that fly first-class. Explain why the Central Limit Theorem cannot be used.

Solution Steps

To determine whether the Central Limit Theorem (CLT) can be applied, we need to check if the sample size is large enough. The rule of thumb for proportions is that both \( n \times p \) and \( n \times (1 - p) \) should be greater than or equal to 10, where \( n \) is the sample size and \( p \) is the population proportion. In this case, we will calculate these values to see if the conditions are met.

We start by calculating \( n \times p \): \[ n \times p = 44 \times 0.09 = 3.96 \]

Next, we calculate \( n \times (1 - p) \): \[ n \times (1 - p) = 44 \times (1 - 0.09) = 44 \times 0.91 = 40.04 \]

For the Central Limit Theorem to be applicable, both \( n \times p \) and \( n \times (1 - p) \) must be greater than or equal to 10. We found:

• \( n \times p = 3.96 \)
• \( n \times (1 - p) = 40.04 \)

Since \( 3.96 < 10 \), the condition is not satisfied.

Final Answer