Distribution of Sample Proportions

Suppose we have a population that is 49% male.  If we take a random sample from that population and determine the percent of our sample that is male, we refer to this proportion as...

• p

• p-hat
  

• the sample mean

• the sampling distribution

When sampling from a population with proportion p, the distribution of sample proportions (p-hat) will vary. What will be the mean of all possible sample proportions?

• p

• $\sqrt{\frac{p\left(1-p\right)}{n}}$

• $\mu$

• $\sigma$

When sampling from a population with proportion p, the sample outcomes will vary.  Which of the following represents the sampling distribution?

• p

• $\sqrt{\frac{p\left(1-p\right)}{n}}$

• $\mu$

• $\sigma$

If certain preconditions are met, we can assume the distribution of all possible p-hats (results of samples taken from the same population) will be normally distributed. Which of the following is NOT a precondition that needs to be met?

• The sample is drawn randomly from the population

• All participants are independent of each other (at least in regards to the characteristic in question)

• $np\ge10$

• $n\left(1-p\right)\ge10$

• $.25\le p\le.75$

Assume we are sampling to find the percent of people of Asian descent from a population that is actually 20% Asian (p=.2). Which of the following sample sizes would meet the minimum requirement for assuming the distribution of sampling proportions is normal? (Hint-remember your pre-conditions). Select ALL that would be considered big enough.

• 10

• 20

• 50

• 75

One of the often overlooked pre-conditions for sampling is the "10% rule." Your sample can't exceed 10% of the population, or else you significantly change the proportion of the population simply by removing your sample. Assume we are sampling from a population of 1700 (Warwick HS). Which of the following sample sizes would not violate the 10% rule? Choose all that qualify.

• 100

• 150

• 200

• 250

Given a population with proportion p=.35. Which of the following is true of a sample of size 50? Assume the sample is taken randomly and all individuals are independent of each other, and we have not violated the 10% rule.

• Pre-conditions for normality have not been met (the sample is too small).

• Sample is big enough, p= .50 and sampling distribution = .35

• Sample is big enough, p= .35 and sampliing distribution = .067

• Sample is big enough, p= .35 and the sampling distribution = .107

Assume we have met all the pre-conditions to ensure our distribution of all possible sample proportions is normal. If I'm drawing a sample of size 100 (n=100) from a population that is 49% men (so p=.49), what is the probability that my sample proportion p-hat will be less than .40? Remember this proportion is normally distributed. 

P(p-hat<.40)=...

• .036

• .452

• .490

• .964

Given a box of 1000 marbles that is 28% red, if I randomly select 36 marbles, have I drawn a large enough sample to use the normal distribution?

• No, because $np<10$

• Yes, because $np\ge10\ and\ n\left(1-p\right)\ge10$

• no, becuase $n\left(1-p\right)\ge10$

Given the same box of marbles with 28% red, if I randomly select 36 marbles what is the probability that I will have at least 9 red marbles in my sample?

In this case p = 9/36 = .25

Hint: remember this is a "greater than or equal" problem

• Sample isn't large enough to use the normal distribution

• P(p-hat>.28) = .50

• P(p-hat>.28) = .66

• P(p-hat>.28) = .72