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%

one or more answer is right but the first situation is wrong.

Explanation / Answer

The rule behind using normal approximation for sampling distribution of proportion is as follows:

For repeated drawing of samples of same size from a population, the sampling distribution (of sample proportions recoreded), can be approximated using normal distribution if, both np and n(1-p) are atleast 10, where, n is sample size, p is probability of success, and 1-p is probability of failure.

Scenario 1: Researcher didnot use random sampling while selecting samples. The sample size, n=42 is not representative of the population of all undergraduate female students at UNC. Therefore, normal approximation can not be applied.

Scenario 2: The sample size, n=60 is representative of all undergraduate students from Appalachian State University aged between 25 and 34. And, np=60*0.053=3.18 and n(1-p)=60*(1-0.053)=56.82. Since, np is not atleast 10, normal approximation cannot be applied.

Scenario 3. The sample size, n=38 is representative of all graduate students at NC State. Both, np=38*0.266=10.108 and n(1-p)=38*(1-0.266)=27.892 are atleast 10. Therefore, normal approximation can be applied.

Sceenario 4. Researcher didnot use random sampling while selecting samples. The sample size, n=20 is not representative of the population of all full time students at Fayetteville State University. Therefore, normal approximation can not be applied.

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