A nutritionist believes that 10% of teenagers eat cereal for breakfast. To investigate this claim, she selects a random sample of 150 teenagers and finds that 25 eat cereal for breakfast. She would like to know if the data provide convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%. Are the conditions for inference met?

A. Yes, the conditions for inference are met.
B. No, the 10% condition is not met.
C. No, the Large Counts Condition is not met.
D. No, the randomness condition is not met.

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vaishnavim26 vaishnavim26 · Answer 2 · 2023-10-23T17:47:50-04:00

All four conditions for inference are met, and the nutritionist can proceed with statistical inference to determine if the data provide convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%.

To determine whether the conditions for inference are met, let's examine each condition individually:

Randomness: The nutritionist selected a random sample of 150 teenagers, ensuring that each teenager had an equal chance of being selected. This satisfies the randomness condition.

Independence: The teenagers are assumed to act independently of each other in their breakfast choices. This assumption is reasonable given that the decision to eat cereal for breakfast is likely influenced by personal preferences and habits rather than the choices of others.

Success/Failure Condition: The sample size (n = 150) is sufficiently large (n ≥ 30) to satisfy the Large Counts Condition. Additionally, the number of successes (x = 25) and failures (n - x = 125) are both greater than 10, meeting the 10% Condition.

Normality Condition: Since the sample size (n = 150) is large enough, the Central Limit Theorem ensures that the sampling distribution of the sample proportion (p) is approximately normal.

Therefore, all four conditions for inference are met, and the nutritionist can proceed with statistical inference to determine if the data provide convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%.