Questions: According to the Transportation Security Administration (TSA) data, 6.5% of the passengers fly first-class. A random sample of size 121 was obtained. Let p̂ be the proportion of the sample that fly first-class. Explain why the Central Limit Theorem cannot be used Select an answer Select an answer the sample is large enough but the sampling technique is biased the Central Limit Theorem can be used only for sample means the sample is large enough but not random the sample is not large enough, i.e. np<10 or n(1-p)<10

Solution Steps

The Central Limit Theorem for proportions states that the sampling distribution of sample proportions will be approximately normal if the following conditions are met:

• Random sample: The sample must be a simple random sample from the population.
• Large sample size: The sample size must be large enough such that _np_ ≥ 10 and _n_(1 - _p_) ≥ 10, where _n_ is the sample size and _p_ is the population proportion.

In this case, we are given that the sample is a random sample and _n_ = 121, and _p_ = 0.065. Now let's check the large sample size condition:

• _np_ = 121 \* 0.065 = 7.865
• _n_(1 - _p_) = 121 \* (1 - 0.065) = 121 \* 0.935 = 113.135

Since _np_ < 10, the large sample size condition is not met.

Final Answer: The Central Limit Theorem cannot be used because the sample size is not large enough (i.e. _np_ < 10 or _n_(1- _p_) < 10).