Asksia AI LOGO

Sia

Question
Math
Posted 7 months ago

In the year 2012,12%2012,12 \% of the residents of the US moved to a different residence. Suppose that several census workers took random samples of n=150n=150 residents from the population and computed the proportion p^\hat{p} of residents in each sample who had moved in 2012.

What would be the shape of the sampling distribution of p^\hat{p} ?
Choose 1 answer:
A) Skewed to the left
B) Skewed to the right
(C) Approximately normal
(D) Uniform
Sign in to unlock the answer
Answer from Sia
Posted 7 months ago
Solution by Steps
step 2
The sample size given is n=150n=150. The population proportion of US residents who moved is p=0.12p=0.12
step 3
According to the Central Limit Theorem, the sampling distribution of the sample proportion p^\hat{p} will be approximately normal if the sample size is large enough
step 4
The rule of thumb for the sample size being large enough is that both npnp and n(1−p)n(1-p) should be greater than 10
step 5
Calculating these values: np=150×0.12=18np = 150 \times 0.12 = 18 and n(1−p)=150×(1−0.12)=150×0.88=132n(1-p) = 150 \times (1-0.12) = 150 \times 0.88 = 132
step 6
Since both npnp and n(1−p)n(1-p) are greater than 10, the sampling distribution of p^\hat{p} will be approximately normal
C
Key Concept
Central Limit Theorem
Explanation
The Central Limit Theorem states that the sampling distribution of the sample mean (or proportion) will be approximately normal if the sample size is large enough, typically if both npnp and n(1−p)n(1-p) are greater than 10.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Unlock Smarter Learning with AskSia Super!

Join Super, our all-in-one AI solution that can greatly improve your learning efficiency.

30% higher accuracy than GPT-4o
Entire learning journey support
The most student-friendly features
Study Other Question