Question

1. Suppose that a population is asked whether they prefer soccer or another sport. Let p= 0.4 denote the proportion of the population that prefers soccer.
   1. Create a graphical summary of the population distribution.
   2. Suppose that a sample of size n=60, drawn from this population, contains 22 individuals that prefer soccer to other sports. Create a graphical summary of this sample data distribution.
   3. verify that the sampling distribution of sample proportions drawn from this population is approximately normal.
   4. Using p=0.4 and n=60, write down the center (“mean”) and the standard deviation of the sampling distribution of sample proportions.
      1. What is the probability that a randomly selected sample of n=60  people from this population will have a sample proportion that prefers soccer which falls between .20 and .35?
      2. What is the probability that a randomly selected sample of n=60 people from this population will have a sample proportion that prefers soccer which falls above .42?
      3. What is the proportion x_R for which 80% of randomly selected samples of n=60 people from this population will have a sample proportion that prefers soccer falling below x_R?
      4. What is the interval [x_L,x_R] centered about the mean in which 55% of randomly selected samples of n=60 people from this population will have a sample proportion that falls within this interval?
      5. What relevance does the approximately normal condition have here?
         

Answer 1

1. Create a graphical summary of the population distribution.

1. Suppose that a sample of size n=60, drawn from this population, contains 22 individuals that prefer soccer to other sports. Create a graphical summary of this sample data distribution.

1. verify that the sampling distribution of sample proportions drawn from this population is approximately normal.

n*p = 60*(22/60) = 22 10

n*(1 - p) = 60*(1 - 22/60) = 38 10

The sampling distribution of sample proportions drawn from this population is approximately normal.

1. Using p=0.4 and n=60, write down the center (“mean”) and the standard deviation of the sampling distribution of sample proportions.

Mean = p = 0.4

Standard deviation = √p(1-p)/n = √0.4(1-0.4)/60 = 0.0632