Which of the following scenarios would it be appropriate to use normal approxima

Question

Which of the following scenarios would it be appropriate to use normal approximation for the sampling distribution of the proportion?

Select all that apply.

Select one or more:

A researcher wishes to find the probability that more than 60% of a sample of undergraduate students from UNC will be female. He samples the first 42 students that walk into the gym on Monday morning. The population proportion of undergraduate females at UNC is known to be 58.9%.

A researcher wishes to find the probability that less than 5% of a sample of undergraduate students from Appalachian State University will be between the ages of 25 and 34. He randomly samples 60 undergraduate students from the student database. The population proportion of undergraduates between the ages of 25 and 34 is 5.3%.

A grad student at NC State wants to know how likely it is that a group of students would be made up of more than 27% graduate students. She will randomly select 38 students and ask them if they are a graduate student or an undergraduate student. The population proportion of grad students at NC State is 26.6%

A full time student at Fayetteville State University wants to know how likely it is that a group of students would be made up of less than 70% full time students. She will ask 20 people that she sees parking in the parking deck if they are full time or part time. The population proportion of full time students at Fayetteville State is known to be 72%

Explanation / Answer

For normal approximation for the sampling distribution of the proportion, sample must be random and np and n(1-p) must be atleast 10.

For first scenario, sample is not random. Therefore, normal approximation can not be applied.

For second scenario, np = 60(0.053) = 3.18 which is less than 10. Therefore, normal approximation can not be applied.

For third scenario, np = 38(0.266) = 10.108, n(1-p) = 38(1-0.266) = 27.892 which are greater than 10 and sample is random. Therefore, normal approximation can be applied.

For fourth scenario, n(1-p) = 20(1-0.72) = 5.6 which is less than 10. Therefore, normal approximation can not be applied.

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